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Internal
Combustion
Engines
Internal Combustion Engines
Applied Thermosciences
Third Edition
Colin R. Ferguson
Allan T. Kirkpatrick
Mechanical Engineering Department
Colorado State University, USA
This edition first published 2016
c 2016, John Wiley & Sons, Ltd
○
First Edition published in 2014
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Library of Congress Cataloging-in-Publication Data
Ferguson, Colin R.
Internal combustion engines : applied thermosciences / Colin R. Ferguson,
Allan T. Kirkpatrick. -- Third edition.
pages cm
Includes bibliographical references and index.
ISBN 978-1-118-53331-4 (hardback)
1. Internal combustion engines. 2. Thermodynamics. I. Ferguson, Colin, R.
II. Kirkpatrick, Allan T. III. Title.
TJ756.F47 2015
621.43--dc23
2015016357
A catalogue record for this book is available from the British Library.
Set in 10/12pt TimesLTStd-Roman by Thomson Digital, Noida, India
1 2016
Contents
Preface xi
Acknowledgments xiii
1. Introduction to Internal Combustion Engines
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Introduction 1
Historical Background 4
Engine Cycles 5
Engine Performance Parameters 9
Engine Configurations 16
Examples of Internal Combustion Engines
Alternative Power Plants 26
References 29
Homework 30
2. Heat Engine Cycles
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
Introduction 32
Constant Volume Heat Addition 33
Constant Pressure Heat Addition 36
Limited Pressure Cycle 37
Miller Cycle 39
Finite Energy Release 41
Ideal Four-Stroke Process and Residual Fraction
Discussion of Gas Cycle Models 62
References 63
Homework 64
54
66
Introduction 66
Thermodynamic Properties of Ideal Gas Mixtures 66
Liquid--Vapor--Gas Mixtures 72
Stoichiometry 76
Low-Temperature Combustion Modeling 79
General Chemical Equilibrium 84
Chemical Equilibrium using Equilibrium Constants 89
References 94
Homework 94
4. Fuel--Air Combustion Processes
4.1
4.2
23
32
3. Fuel, Air, and Combustion Thermodynamics
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
1
97
Introduction 97
Combustion and the First Law
97
v
vi
Contents
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
Maximum Work and the Second Law 103
Fuel--Air Otto Cycle 108
Four-Stroke Fuel--Air Otto Cycle 113
Homogeneous Two-Zone Finite Heat Release Cycle 116
Comparison of Fuel--Air Cycles with Actual Spark Ignition Cycles
Limited Pressure Fuel--Air Cycle 125
Comparison of Limited Pressure Fuel--Air Cycles with Actual
Compression Ignition Cycles 128
References 129
Homework 129
5. Intake and Exhaust Flow
5.1
5.2
5.3
5.4
5.5
5.6
5.7
131
Introduction 131
Valve Flow 131
Intake and Exhaust Flow 147
Superchargers and Turbochargers 150
Effect of Ambient Conditions on Engine and Compressor
Mass Flow 158
References 159
Homework 160
6. Fuel and Airflow in the Cylinder
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
Introduction 163
Carburetion 163
Fuel Injection--Spark Ignition 166
Fuel Injection--Compression Ignition 168
Large-Scale in-Cylinder Flow 174
In-Cylinder Turbulence 180
Airflow in Two-Stroke Engines 185
References 193
Homework 195
7. Combustion Processes in Engines
7.1
7.2
7.3
7.4
7.5
7.6
7.7
197
Introduction 197
Combustion in Spark Ignition Engines 198
Abnormal Combustion (Knock) in Spark Ignition Engines
Combustion in Compression Ignition Engines 214
Low-Temperature Combustion 225
References 229
Homework 231
8. Emissions
8.1
8.2
8.3
8.4
8.5
163
234
Introduction 234
Nitrogen Oxides 235
Carbon Monoxide 243
Hydrocarbons 245
Particulates 249
206
123
Contents
8.6
8.7
8.8
9. Fuels
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
Emissions Regulation and Control
References 258
Homework 259
262
Introduction 262
Hydrocarbon Chemistry 263
Refining 266
Fuel Properties 267
Gasoline Fuels 269
Alternative Fuels for Spark Ignition Engines
Hydrogen 281
Diesel Fuels 282
References 286
Homework 287
10. Friction and Lubrication
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10
10.11
10.12
10.13
10.14
288
318
Introduction 318
Engine Cooling Systems 319
Engine Energy Balance 320
Cylinder Heat Transfer 324
Heat Transfer Modeling 326
Heat Transfer Correlations 330
Heat Transfer in the Exhaust System
Radiation Heat Transfer 339
Mass Loss or Blowby 340
References 342
Homework 344
12. Engine Testing and Control
12.1
12.2
274
Introduction 288
Friction Coefficient 288
Friction Mean Effective Pressure 291
Friction Measurements 291
Friction Modeling 294
Journal Bearing Friction 295
Piston and Ring Friction 298
Valve Train Friction 306
Accessory Friction 308
Pumping Mean Effective Pressure 310
Overall Engine Friction Mean Effective Pressure 311
Lubrication 312
References 315
Homework 316
11. Heat and Mass Transfer
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
11.10
11.11
251
Introduction 346
Instrumentation 347
346
338
vii
viii
Contents
12.3
12.4
12.5
12.6
12.7
12.8
Combustion Analysis 354
Exhaust Gas Analysis 358
Control Systems in Engines 366
Vehicle Emissions Testing 369
References 370
Homework 370
13. Overall Engine Performance
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
13.10
Appendices
A
B
C
D
E
F
372
Introduction 372
Effect of Engine and Piston Speed 372
Effect of Air--Fuel Ratio and Load 373
Engine Performance Maps 376
Effect of Engine Size 379
Effect of Ignition and Injection Timing 380
Effect of Compression Ratio 383
Vehicle Performance Simulation 383
References 384
Homework 385
387
Physical Properties of Air 387
Thermodynamic Property Tables for Various Ideal Gases 389
Curve-Fit Coefficients for Thermodynamic Properties of Various Fuels and
Ideal Gases 397
Conversion Factors and Physical Constants 401
Thermodynamic Analysis of Mixtures 403
E.1 Thermodynamic Derivatives 403
E.2 Numerical Solution of Equilibrium Combustion Equations 405
E.3 Isentropic Compression/Expansion with Known Δ𝑃 408
E.4 Isentropic Compression/Expansion with Known Δ𝑣 409
E.5 Constant Volume Combustion 410
E.6 Quality of Exhaust Products 411
E.7 References 412
Computer Programs 413
F.1 Volume.m 414
F.2 Velocity.m 414
F.3 BurnFraction.m 414
F.4 FiniteHeatRelease.m 415
F.5 FiniteHeatMassLoss.m 417
F.6 FourStrokeOtto.m 420
F.7 RunFarg.m 421
F.8 farg.m 422
F.9 fuel.m 425
F.10 RunEcp.m 426
F.11 ecp.m 427
F.12 AdiabaticFlameTemp.m 437
F.13 OttoFuel.m 438
Contents
F.14
F.15
F.16
F.17
Index
455
FourStrokeFuelAir.m 440
Homogeneous.m 444
Friction.m 450
WoschniHeatTransfer.m 451
ix
Preface
This textbook presents a modern approach to the study of internal combustion engines.
Internal combustion engines have been, and will remain for the foreseeable future, a vital
and active area of engineering education and research. The purpose of this book is to apply
the principles of thermodynamics, fluid mechanics, and heat transfer to the analysis of
internal combustion engines. This book is intended first to demonstrate to the student the
application of engineering sciences, especially the thermal sciences, and second, it is a book
about internal combustion engines. Considerable effort is expended making the requisite
thermodynamics accessible to students. This is because most students have little, if any,
experience applying the first law to unsteady processes in open systems or in differential
form to closed systems, and have experience with only the simplest of reacting gas mixtures.
The text is designed for a one-semester course in internal combustion engines at the
senior undergraduate level. At Colorado State University, this text is used for a single term
class in internal combustion engines. The class meets for a lecture two times per week and
a recitation/laboratory once a week, for a term of 15 weeks.
This third edition builds upon the foundation of the second edition. The major changes
are the adoption of the programming software MATLABⓇ for the examples, and chapter
reorganization for a greater emphasis on combustion. The content changes include additional topics on heat and mass loss in finite heat release models, thermodynamic properties
of reacting mixtures, two-zone burn models for homogeneous mixtures, exhaust blowdown
modeling, diesel fuel injection, NO𝑥 concentration using finite rate chemistry, homogeneous charge compression ignition, and alternative fuels. The homework problems have
increased in number and topics covered.
xi
Acknowledgments
The approach and style of this text reflects our experiences as students at the Massachusetts
Institute of Technology. In particular, we learned a great deal from MIT Professors John
B. Heywood, Warren M. Rohsenow, Ascher Shapiro, and Jean F. Louis.
Many thanks to the editorial staff at John Wiley & Sons for their work on the third
edition. Mr. Paul Petralia, Mr. Clive Lawson, Ms. Sandra Grayson, and Ms. Shikha Pahuja
deserve special acknowledgement for their editorial assistance with this project. This edition
also benefited from technical discussions with Professors Anthony Marchese, Daniel Olsen,
and Brian Willson. Mr. Aron Dobos, a CSU ME graduate student, deserves thanks for
converting many of the computer programs in the first and second editions to a MatlabⓇ
form. Mr. Tyler Schott helped produce and format the solutions to the homework problems.
Finally, Allan Kirkpatrick would like to thank his family: Susan, Anne, Matt, Rob, and
Kristin for their unflagging support while this third edition was being written.
Dr. Allan T. Kirkpatrick (allan@engr.colostate.edu)
Fort Collins, Colorado
xiii
Chapter
1
Introduction to Internal
Combustion Engines
1.1 INTRODUCTION
The main focus of this text is on the application of the engineering sciences, especially the
thermal sciences, to internal combustion engines. The goals of the text are to familiarize
the reader with engine nomenclature, describe how internal combustion engines work, and
provide insight into how engine performance can be modeled and analyzed. An internal
combustion engine is defined as an engine in which the chemical energy of the fuel is
released inside the engine and used directly for mechanical work, as opposed to an external
combustion engine in which a separate combustor is used to burn the fuel.
In this chapter, we discuss the engineering parameters that are used to characterize
the overall performance of internal combustion engines. Major engine cycles, configurations, and geometries are covered. The following chapters will apply the principles of
thermodynamics, combustion, fluid flow, friction, and heat transfer to determine an internal combustion engine’s temperature and pressure profiles, work, thermal efficiency, and
exhaust emissions.
An aspect upon which we have put considerable emphasis is the process of constructing
idealized models to represent actual physical situations in an engine. Throughout the text, we
will calculate the values of the various thermal and mechanical parameters that characterize
internal combustion engine operation.
With the advent of high-speed computers and advanced measurement techniques,
today’s internal combustion engine design process has evolved from being purely empirical
to a rigorous semiempirical process in which computer-based engineering software is used
to evaluate the performance of a proposed engine design even before the engine is built
and tested. The development of a successful engine requires knowledge of methods and
analyses introduced in the text which are used to parameterize and correlate experiments,
and to calculate the performance of a proposed engine design.
The internal combustion engine was invented and successfully developed in the late
1860s. It is considered as one of the most significant inventions of the last century, and has
had a significant impact on society, especially human mobility. The internal combustion
engine has been the foundation for the successful development of many commercial technologies. For example, consider how the internal combustion engine has transformed the
transportation industry, allowing the invention and improvement of automobiles, trucks,
airplanes, and trains. The adoption and continued use of the internal combustion engine
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
1
2
Introduction to Internal Combustion Engines
Figure 1.1 Piston and connecting rod.
(Courtesy Mahle, Inc.)
in different application areas has resulted from its relatively low cost, favorable power to
weight ratio, high efficiency, and relatively simple and robust operating characteristics.
The reciprocating piston--cylinder geometry is the primary geometry that has been
used in internal combustion engines, and is shown in Figure 1.1. As indicated in the
figure, a piston oscillates back and forth in a cyclic pattern in a cylinder, transmitting
power to a drive shaft through a connecting rod and crankshaft mechanism. Valves or
ports are used to control the flow of gas into and out of the engine. This configuration of a
reciprocating internal combustion engine, with an engine block, pistons, valves, crankshaft,
and connecting rod, has remained basically unchanged since the late 1800s.
The main differences between a modern-day engine and one built 100 years ago can be
seen by comparison of their reliability, thermal efficiency, and emissions level. For many
years, internal combustion engine research was aimed at improving thermal efficiency and
reducing noise and vibration. As a consequence, the thermal efficiency has increased from
about 10--20% at the beginning of the 20th century to values as high as 50% today.
Internal combustion engine efficiency continues to increase, driven both by legislation
and the need to reduce operating costs. The primary United States vehicle mileage standard
is the federal corporate average fuel economy (CAFE) standard. The CAFE standard for
passenger vehicles and light duty trucks was 27.5 miles per gallon (mpg) for a 20 year period
from 1990 to 2010. The CAFE standards have risen in the last few years, and will reach
35.5 mpg in 2016, and 54.5 mpg by 2025. This doubling of vehicle mileage requirements
will require increased use of techniques such as electronic control, engine downsizing,
turbocharging, supercharging, variable valve timing, low temperature combustion, and
electric motors and transmissions.
Internal combustion engines have become the dominant prime mover technology in
several areas. For example, in 1900 most automobiles were steam or electrically powered,
but by 1920 most automobiles were powered by gasoline engines. As of the year 2010,
in the United States alone there are about 220 million motor vehicles powered by internal
combustion engines. In 1900, steam engines were used to power ships and railroad locomotives; today two- and four-stroke diesel engines are used. Prior to 1950, aircraft relied
Introduction
3
Figure 1.2 Automobile engine. (Courtesy
Mercedes-Benz Photo Library.)
almost exclusively on piston engines. Today gas turbines are the power plant used in large
planes, and piston engines continue to dominate the market in small planes.
Internal combustion engines have been designed and built to deliver power in the range
from 0.01 to 20 × 103 kW, depending on their displacement. They compete in the market
place with electric motors, gas turbines, and steam engines. The major applications are in
the vehicular (see Figure 1.2), railroad, marine (see Figure 1.3), aircraft, stationary power,
and home use areas. The vast majority of internal combustion engines are produced for
vehicular applications, requiring a power output on the order of 100 kW.
Since 1970, with the recognition of the importance of environmental issues such as the
impact of air quality on health, there has also been a great deal of work devoted to reducing
the various emissions from engines. The emissions level of current internal combustion
engines has decreased to about 5% of the emissions levels 40 years ago. Currently, meeting
emission requirements is one of the major factors in the design and operation of internal combustion engines. The major emissions from internal combustion engines include
Figure 1.3 Marine engine. (Courtesy Man
B&W Diesel.)
4
Introduction to Internal Combustion Engines
nitrogen oxides (NO𝑥 ), carbon monoxide (CO), hydrocarbons (HC), particulates (PM), and
aldehydes. These combustion products are a significant source of air pollution, as the internal combustion engine is the source of about half of the NO𝑥 , CO, and HC pollutants in the
air. Carbon dioxide (CO2 ), a primary gaseous combustion product of internal combustion
engines is also a greenhouse gas, and is in the process of being regulated as well.
1.2 HISTORICAL BACKGROUND
In this section, we briefly discuss a few of the major figures in the invention and development
of the internal combustion engine. The ingenuity and creativity demonstrated by these
early engineers in producing these successful inventions is truly inspiring to today’s engine
designers. In 1858, J. Lenior (1822--1900), a Belgian engineer, developed a two-stroke
engine that developed 6 hp with an efficiency of about 5%. During the intake stroke, a
gas--air mixture at atmospheric pressure was drawn into the engine, and ignited by a spark,
causing the cylinder pressure to increase during the latter half of the stroke, producing
work. The return stroke was used to remove the combustion products through an exhaust
valve. The Lenior engine was primarily used in stationary power applications.
In 1872, George Brayton (1830--1892), an American mechanical engineer, patented
and commercialized a constant pressure internal combustion engine, ‘‘Brayton’s Ready Engine’’. The engine used two reciprocating piston-driven cylinders, a compression cylinder,
and an expansion cylinder. This cycle was also called the ‘‘flame cycle’’, as ignition of the
gas--air mixture was by a pilot flame, and the mixture was ignited and burned at constant
pressure as it was pumped from the compression cylinder to the expansion cylinder. The
Brayton piston engine was used on the first automobile in 1878. The Brayton cycle is the
thermodynamic cycle now used by gas turbines, which use rotating fan blades to compress
and expand the gas flowing through the turbine.
Nikolaus Otto (1832--1891), a German engineer, developed the ‘‘Otto Silent Engine’’,
the first practical four-stroke engine with in-cylinder compression in 1876. With a compression ratio of 2.5, the gas engine produced 2 hp at 160 rpm, and had a brake efficiency of
14%. Nikolaus Otto is considered the inventor of the modern internal combustion engine,
and the founder of the internal combustion engine industry. The concept of a four-stroke
engine had been conceived and patented by A. de Rochas in 1861, however Otto is recognized as the first person to build and commercialize a working flame ignition engine. Otto
had no formal engineering schooling, and was self-taught. He devoted his entire career to
the advancement of the internal combustion engine. In 1872, he founded the first internal combustion engine manufacturing company, N. A. Otto and Cie, and hired Gottlieb
Daimler and Wilhelm Maybach, who would go on to start the first automobile company,
the Daimler Motor Company in 1890. Otto’s son Gustav founded the automotive company
now known as BMW.
The first practical two-stroke engine was invented and built by Sir Dugald Clerk
(1854--1932), a Scottish mechanical engineer, in 1878. Clerk graduated from Yorkshire
College in 1876, and patented his two-stroke engine in 1881. He is well known for his
career-long contributions to improvement of combustion processes in large-bore two-stroke
engines. Clerk’s engine was made of two cylinders--one a working cylinder to produce
power, and the other a pumping cylinder to compress and transfer the intake air and fuel
mixture to the working cylinder. Poppet valves were used for intake flow, and a cylinder port
uncovered by the piston on the expansion stroke was used to exhaust the combustion gases.
Many of these early internal combustion engines, such as the Lenior, Brayton, and Otto
engines, were powered by coal gas, a mixture of methane, hydrogen, carbon monoxide, and
Engine Cycles
5
other gases produced by the partial pyrolysis of coal. In the 1880s, crude oil refineries began
producing gasoline and kerosene in quantities sufficient to create a market for liquid-fueled
internal combustion engines.
Gottlieb Daimler (1834--1900), a German engineer, is recognized as one of the founders
of the automotive industry. He developed a high-speed four-stroke gasoline-fueled engine
in 1883. The liquid fuel was vaporized and mixed with the intake air in a carburetor before
being drawn into the combustion chamber. The fuel air mixture was ignited by a flame
tube. In 1886, he built the first four-wheeled automobile, and founded the Daimler Motor
Company in 1890.
Karl Benz (1844--1929), a German engineer, successfully developed a 3.5 hp liquidfueled two-stroke engine with a carburetor and spark ignition in 1885. The ignition system
consisted of an electrical induction coil with a rotary breaker driven by the engine and a removable spark plug fitted into the cylinder head, similar to what is found in today’s engines.
The engine was installed on a three-wheeled vehicle in 1886, the first ‘‘horseless carriage’’.
The transmission was a two-chain arrangement that connected the engine to the rear axle.
In 1897, Rudolph Diesel (1858--1913), a German engineer, developed the first practical
four-stroke engine using direct injection of liquid fuel into the combustion chamber. The
high compression ratio of the engine resulted in autoignition and combustion of the fuel air
mixture. Diesel graduated from Munich Polytechnic in 1880, and worked with his former
professor, Carl von Linde, initially on ammonia Rankine cycle refrigeration, then worked
with the MAN company to develop compression ignition engines. He designed his engines
to follow Carnot’s thermodynamic principles as closely as possible. Accordingly, his initial
objective was to have constant temperature combustion, however, this was not realized in
practice, and he adopted the strategy of constant pressure combustion.
Rudolph Diesel’s single-cylinder engine had a bore of 250 mm, stroke of 400 mm, for
a 20 L displacement. The diesel fuel was atomized using air injection, a technique where
compressed air entrained diesel fuel in the injector and carried it into the cylinder. The
engine operated at a speed of 170 rpm, and produced 18 hp, with an efficiency of 27%
at full load. This is a much greater efficiency than the steam engines and spark ignition
engines in use at that time.
Sir Harry Ricardo (1885--1974), a mechanical engineering graduate of Cambridge,
and a prominent English engineer, patented the use of a spherical prechamber, the Ricardo
‘‘Comet’’, to greatly increase the fuel--air mixing rate, allowing diesel engines to be used in
high--speed, 2000 rpm and higher, engine vehicular applications. The first multi-cylinder
diesel engines for trucks were available by 1924, and the first diesel-powered automobiles
were available by 1936. During his career, Ricardo also contributed to greater understanding
of the role of turbulence, swirl and squish in enhancing flame speed in both spark and
diesel engines, commercialized sleeve valves for aircraft engines, developed an octane
rating system for quantifying knock in spark engines, and founded what is now the Ricardo
Consulting Engineers Company.
These early engines were air cooled, since they produced relatively low power. Naturalconvection water-cooling using the thermosyphon principle, and forced convection cooling
using water pumps was adopted after about 1910 for higher horsepower engines. For
example, Henry Ford’s Model T engine of 1908, and the Wright Brother’s Flyer engine of
1903 used natural convection water cooling.
1.3 ENGINE CYCLES
The two major cycles currently used in internal combustion engines are termed Otto and
Diesel, named after the two men credited with their invention. The Otto cycle is also
6
Introduction to Internal Combustion Engines
known as a constant volume combustion or spark ignition cycle, and the Diesel cycle is
also known as a constant pressure combustion or compression ignition cycle. These cycles
can configured as either a two-stroke cycle in which the piston produces power on every
downward stroke, or a four-stroke cycle in which the piston produces power every other
downward stroke.
Otto Cycle
As shown in Figure 1.4, the four-stroke Otto cycle has the following sequence of operations:
1. An intake stroke that draws a combustible mixture of fuel and air past the throttle and
the intake valve into the cylinder.
2. A compression stroke with the valves closed that raises the temperature of the mixture.
A spark ignites the mixture toward the end of the compression stroke.
3. An expansion or power stroke resulting from combustion of the fuel--air mixture.
4. An exhaust stroke that pushes out the burned gases past the exhaust valve.
Intake
Spark
plug
Cylinder
Piston
Crankshaft
Compression
Intake
Exhaust
port
Intake
port
Power
Figure 1.4 Four-stroke spark ignition cycle.
Exhaust
Exhaust
Engine Cycles
7
Air enters the engine through the intake manifold, a bundle of passages that evenly
distribute the air mixture to individual cylinders. The fuel, typically gasoline, is mixed with
the inlet air using a fuel injector or carburetor in the intake manifold, intake port, or directly
injected into the cylinder, resulting in the cylinder filling with a homogeneous mixture.
When the mixture is ignited by a spark, a turbulent flame develops and propagates through
the mixture, raising the cylinder temperature and pressure. The flame is extinguished when
it reaches the cylinder walls. If the initial pressure is too high, the compressed gases ahead
of the flame will autoignite, causing a problem called knock. The occurrence of knock
limits the maximum compression ratio and thus the efficiency of Otto cycle engines. The
burned gases exit the engine past the exhaust valves through the exhaust manifold. The
exhaust manifold channels the exhaust from individual cylinders into a central exhaust
pipe.
In the Otto cycle, a throttle is used to control the amount of air inducted. As the throttle
is closed, the amount of air entering the cylinder is reduced, causing a proportional reduction
in the cylinder pressure. Since the fuel flow is metered in proportion to the airflow, the
throttle in an Otto cycle, in essence, controls the power.
Diesel Cycle
The four-stroke Diesel cycle has the following sequence:
1. An intake stroke that draws inlet air past the intake valve into the cylinder.
2. A compression stroke that raises the air temperature above the autoignition temperature
of the fuel. Diesel fuel is sprayed into the cylinder near the end of the compression
stroke.
3. Evaporation, mixing, ignition, and combustion of the diesel fuel during the later stages
of the compression stroke and the expansion stroke.
4. An exhaust stroke that pushes out the burned gases past the exhaust valve.
There are two types of diesel combustion systems, direct injection (DI) into the main
cylinder, and indirect injection (IDI) into a prechamber connected to the main cylinder.
With indirect injection, air is compressed into a prechamber during the compression stroke,
producing a highly turbulent flow field, and thus high mixing rates when the diesel fuel
is sprayed into the prechamber toward the end of the compression stroke. The combustion
process is initiated in the prechamber, raising the pressure in the prechamber above that
of the main chamber, which forces the combusting mixture of burning gases, fuel, and air
back into the main chamber, resulting in the propagation of a highly turbulent swirling
flame into the main chamber. Indirect injection engines tend to be used where the engine
is expected to perform over a wide range of speeds and loads such as in an automobile.
When the operating range of the engine is less broad such as in ships, trucks, locomotives,
or electric power generation, direct injection engines predominate.
The inlet air in the diesel engine is unthrottled, and the combustion is lean. The power
is controlled by the amount of fuel injected and the subsequent mixing of the fuel spray
with the inlet air. The injection duration is proportional to the engine load. In order to
ignite the fuel--air mixture, diesel engines are required to operate at a higher compression
ratio, compared to spark ignition (SI) engines, with typical values in the range of 15--20,
resulting in a greater theoretical efficiency. Since the diesel fuel is mixed with cylinder air
just before combustion is to commence, the knock limitation that occurs in SI engines is
greatly reduced.
8
Introduction to Internal Combustion Engines
Diesel engine performance is limited by the time required to mix the fuel and air,
as incomplete mixing and combustion results in decreased power, increased unburned
hydrocarbon emissions, and visible smoke. As we shall see, many different diesel combustion chamber designs have been invented to achieve adequate mixing. Since the mixing
time is inversely proportional to the engine speed, diesel engines are classified into three
classes, high-speed, medium speed, and low speed. High-speed diesels are designed to
operate at speeds of 1000 rpm or higher, have up to a 300 mm bore, and use high-quality
distillate fuels. Medium-speed diesels operate at speeds ranging from 375 to 1000 rpm,
have a medium bore typically between 200 and 600 mm, and can operate with a range
of fuels. The low-speed class of diesel engines operate at speeds less than 375 rpm, are
typically large-bore (> 600 mm) two-stroke cycle engines, and use residual fuel oil. Each
engine manufacturer has worked to optimize the design for a particular application, and
that each manufacturer has produced an engine with unique characteristics illustrates that
the optimum design is highly dependent on the specific application.
Two-Stroke Cycle
As the name implies, two-stroke engines need only two-strokes of the piston or one revolution to complete a cycle. There is a power stroke every revolution instead of every two
revolutions as for four-stroke engines. Two-stroke engines are mechanically simpler than
four-stroke engines, and have a higher specific power, the power to weight ratio. They
can use either spark or compression ignition cycles. One of the performance limitations of
two-stroke engines is the scavenging process, simultaneously exhausting the burnt mixture
and introducing the fresh fuel--air mixture into the cylinder. As we shall see, a wide variety
of two-stroke engines have been invented to ensure an acceptable level of scavenging.
The principle of operation of a crankcase-scavenged two-stroke engine, developed by
Joseph Day (1855--1946), is illustrated in Figure 1.5. During compression of the crankcasescavenged two-stroke cycle, a subatmospheric pressure is created in the crankcase. In the
example shown, this opens a reed valve letting air rush into the crankcase. Once the piston
reverses direction during combustion and expansion begins, the air in the crankcase closes
Spark plug
(or fuel injector)
Exhaust
ports
Intake
ports
Reed
valve
Fuel–air
(or air)
• Compression
• Ports closed
• Air inducted
into crankcase
• Combustion, expansion
• Ports closed
• Exhaust
• Intake port closed
Air compressed in crankcase
(Reed valve shut)
Figure 1.5 A cross-scavenged two-stroke cycle.
• Scavenging
• Intake
• Ports open
• Reed valve shut
Engine Performance Parameters
9
the reed valve so that the air is compressed. As the piston travels further, it uncovers holes
or exhaust ports, and exhaust gases begin to leave, rapidly dropping the cylinder pressure
to that of the atmosphere. Then the intake ports are opened and compressed air from the
crankcase flows into the cylinder pushing out the remaining exhaust gases. This pushing
out of exhaust by the incoming air is called scavenging.
Herein lies one problem with two-stroke engines: the scavenging is not perfect; some of
the air will go straight through the cylinder and out the exhaust port, a process called shortcircuiting. Some of the air will also mix with exhaust gases and the remaining incoming air
will push out a portion of this mixture. The magnitude of the problem is strongly dependent
on the port designs and the shape of the piston top.
Less than perfect scavenging is of particular concern if the engine is a carbureted
gasoline engine, for instead of air being in the crankcase there is a fuel--air mixture.
Some of this fuel--air mixture will short circuit and appear in the exhaust, wasting fuel
and increasing the hydrocarbon emissions. Carbureted two-stroke engines are used where
efficiency is not of primary concern and advantage can be taken of the engine’s simplicity;
this translates into lower cost and higher power per unit weight. Familiar examples include
motorcycles, chain saws, outboard motors, and model airplane engines. However, use in
motorcycles is decreasing because they have poor emission characteristics. Two-stroke
industrial engines are mostly diesel, and typically supercharged. With a two-stroke diesel
or fuel injected gasoline engine, air only is used for scavenging, so loss of fuel through
short-circuiting or mixing with exhaust gases is not a problem.
1.4 ENGINE PERFORMANCE PARAMETERS
Engine Geometry
For any one cylinder, the crankshaft, connecting rod, piston, and head assembly can be
represented by the mechanism shown in Figure 1.6. Of particular interest are the following
geometric parameters: bore, 𝑏; connecting rod length, 𝑙; crank radius, 𝑎; stroke, 𝑠; and crank
angle, 𝜃. The crank radius is one-half of the stroke. The top dead center (tdc) of an engine
b
tdc
y
s
Piston
bdc
l
Connecting
rod
a
Crankshaft
Figure 1.6 Engine slider--crank geometry.
10
Introduction to Internal Combustion Engines
refers to the crankshaft being in a position such that 𝜃 = 0◦ . The cylinder volume in this
position is minimum and is also called the clearance volume, 𝑉c . Bottom dead center (bdc)
refers to the crankshaft being at 𝜃 = 180◦ . The cylinder volume at bottom dead center 𝑉1
is maximum.
The compression ratio, 𝑟, is defined as the ratio of the maximum to minimum volume.
𝑟=
𝑉bdc
𝑉
= 1
𝑉tdc
𝑉c
(1.1)
The displacement volume, 𝑉d , is the difference between the maximum and minimum
volume; for a single cylinder,
𝜋
(1.2)
𝑉d = 𝑉1 − 𝑉c = 𝑏2 𝑠
4
A useful expression relating 𝑉d and 𝑉bdc is
𝑟
𝑉1 = 𝑉bdc =
(1.3)
𝑉
𝑟−1 d
For multicylinder engines, the total displacement volume is the product of the number of
cylinders, 𝑛c , and the volume of a single cylinder.
𝜋
𝑉d = 𝑛c 𝑏2 𝑠
(1.4)
4
The mean piston speed 𝑈̄ p is an important parameter in engine design since stresses and
other factors scale with piston speed rather than with engine speed. Since the piston travels
a distance of twice the stroke per revolution, it should be clear that
𝑈̄ p = 2𝑁𝑠
(1.5)
The engine speed, 𝑁, refers to the rotational speed of the crankshaft and is expressed
in revolutions per minute. The engine frequency, 𝜔, also refers to the rotation rate of the
crankshaft but in units of radians per second.
Power, Torque, and Efficiency
The brake power, 𝑊̇ b , is the rate at which work is done; and the engine torque, 𝜏, is a
measure of the work done per unit rotation (radians) of the crank. The brake power is the
power output of the engine, and measured by a dynamometer. Early dynamometers were
simple brake mechanisms. The brake power is less than the boundary rate of work done by
the gas, called indicated power, partly because of friction. As we shall see when discussing
dynamometers in Chapter 10, the brake power and torque are related by
𝑊̇ b = 2𝜋𝜏𝑁
(1.6)
The net power is from the complete engine, whereas gross power is from an engine
without the cooling fan, muffler, and tail pipe.
The indicated work 𝑊i is the net work transferred from the gas to the piston during a
cycle, which is the integral of the pressure over the cylinder volume:
𝑊i =
∫
𝑃 𝑑𝑉
(1.7)
and the indicated power 𝑊̇ i , for an engine with 𝑛c cylinders, is
𝑊̇ i = 𝑛c 𝑊i 𝑁∕2 (four stroke engine)
𝑊̇ i = 𝑛c 𝑊i 𝑁
(two stroke engine)
(1.8)
(1.9)
Engine Performance Parameters
60
50
90
80
70
60
Power
70
11
40
50
(lb ft)
110
(Nm)
150
30
(kW)
40
(hp)
Brake torque
140
100
130
90
80
Figure 1.7 Wide open throttle
(WOT) performance of an automotive four-stroke engine.
120
110
1000
2000 3000 4000 5000
Engine speed (rpm)
6000
since the four-stroke engine has two revolutions per power stroke and the two-stroke engine
has one revolution per power stroke.
The brake power is less than the indicated power due to engine mechanical friction,
pumping losses in the intake and exhaust, and accessory power needs, which are grouped
as a friction power loss, 𝑊̇ f
𝑊̇ f = 𝑊̇ i − 𝑊̇ b
(1.10)
The ratio of the brake power to the indicated power is the mechanical efficiency, 𝜂m :
𝜂m = 𝑊̇ b ∕𝑊̇ i = 1 − 𝑊̇ f ∕𝑊̇ i
(1.11)
The wide open throttle performance of a 2.0 L automotive four-stroke engine is plotted
in Figure 1.7. As with most engines, the torque and power both exhibit maxima with engine
speed. Viscous friction effects increase quadratically with engine speed, causing the torque
curve to decrease at high engine speeds. The maximum torque occurs at lower speed than
maximum power, since power is the product of torque and speed. Notice that the torque
curve is rippled. This is due to both inlet and exhaust airflow dynamics and mechanical
friction, discussed later.
Mean Effective Pressure
The mean effective pressure (mep) is the work done per unit displacement volume, and
has units of force/area. It is the average pressure that results in the same amount of work
actually produced by the engine. The mean effective pressure is a very useful parameter
as it scales out the effect of engine size, allowing performance comparison of engines of
different displacement. There are three useful mean effective pressure parameters--mep,
bmep, and fmep.
12
Introduction to Internal Combustion Engines
The indicated mean effective pressure (imep) is the net work per unit displacement
volume done by the gas during compression and expansion. The name originates from the
use of an ‘‘indicator’’ card used to plot measured pressure versus volume. The pressure in
the cylinder initially increases during the expansion stroke due to the heat addition from
the fuel, and then decreases due to the increase in cylinder volume.
The brake mean effective pressure (bmep) is the external shaft work per unit volume
done by the engine. The name originates from the ‘‘brake’’ dynamometer used to measure
the torque produced by the rotating shaft. Typical values of measured bmep for naturally
aspirated automobile engines depend on the load, with maximum values of about 10 bar,
and greater values of about 20 bar for turbo or supercharged engines.
Based on torque, the bmep is
bmep =
4𝜋𝜏
𝑉d
(four stroke engine)
=
2𝜋𝜏
𝑉d
(two stroke engine)
(1.12)
and in terms of power the bmep is
bmep =
=
𝑊̇ b
𝑉d 𝑁∕2
𝑊̇ b
𝑉d 𝑁
(four stroke engine)
(1.13)
(two stroke engine)
The bmep can also be expressed in terms of piston area 𝐴p , mean piston speed 𝑈̄ p , and
number of cylinders 𝑛c :
bmep =
4𝑊̇ b
𝑛c 𝐴p 𝑈̄ p
(four stroke engine)
=
2𝑊̇ b
𝑛c 𝐴p 𝑈̄ p
(two stroke engine)
(1.14)
The friction mean effective pressure (fmep) includes the mechanical engine friction,
the pumping losses during the intake and exhaust strokes, and the work to run auxiliary
components such as oil and water pumps. Accordingly, the friction mean effective pressure
(fmep) is the difference between the imep and the bmep. Determination of the fmep is
discussed further in Chapter 10.
f mep = imep − bmep
(1.15)
The bmep of two different displacement automobile engines at wide open throttle
(WOT) is compared versus mean piston speed in Figure 1.8. Notice that when performance
is scaled to be size independent, there is considerable similarity.
Volumetric Efficiency
A performance parameter of importance for four-stroke engines is the volumetric efficiency,
𝑒v . It is defined as the mass of fuel and air inducted into the cylinder divided by the mass
that would occupy the displaced volume at the density 𝜌i in the intake manifold. The flow
restrictions in the intake system, including the throttle, intake port, and valve, create a
pressure drop in the inlet flow, which reduces the density and thus the mass of the gas in the
Engine Performance Parameters
13
12
bmep (bar)
10
8
2.0 L
6
3.8 L
4
Figure 1.8 Brake mean
effective pressure at
WOT versus mean piston
speed for two automotive
engines.
2
0
2
4
6
8
10
12
14
Mean piston speed (m/s)
16
18
20
cylinder. The volumetric efficiency is a mass ratio and not a volume ratio. The volumetric
efficiency for an engine operating at a speed 𝑁 is thus
𝑒v =
𝑚̇ in
𝜌i 𝑉d 𝑁∕2
(1.16)
where
𝑚̇ in = 𝑚̇ a + 𝑚̇ f
(1.17)
In Equation 1.17, 𝑚̇ f is the flow rate of the fuel inducted in the intake manifold. For
a direct injection engine, 𝑚̇ f = 0. The factor of 2 accounts for the two revolutions per
cycle in a four-stroke engine. The intake manifold density is used as a reference condition
instead of the standard atmosphere, so that supercharger performance is not included in the
definition of volumetric efficiency. For two-stroke cycles, a parameter related to volumetric
efficiency called the delivery ratio is defined in terms of the airflow only and the ambient
air density instead of the intake manifold density.
A representative plot of volumetric efficiency versus engine speed of an automotive
four-stroke engine is shown in Figure 1.9. The shape and location of the peaks of the
volumetric efficiency curve are very sensitive to the engine speed as well as the manifold
configuration. Some configurations produce a flat curve, others produce a very peaked and
asymmetric curve. As we will see later, the volumetric efficiency is also influenced by the
Volume efficiency (%)
90
Equal spacing, single plane
Equal spacing, over/under
80
Unequal spacing, individual
runners, octopus, single plane
70
2000 3000 4000 5000
Engine speed (rpm)
Figure 1.9 Effect of engine speed and intake manifold geometry on volumetric efficiency.
Adapted from Armstrong and Stirrat (1982).
14
Introduction to Internal Combustion Engines
valve size, valve lift, and valve timing. It is desirable to maximize the volumetric efficiency
of an engine, since the amount of fuel that can be burned and power produced for a given
engine displacement (hence size and weight) is maximized. Although it does not influence
in any way the thermal efficiency of the engine, the volumetric efficiency will influence
the overall thermal efficiency of the system in which it is installed. As Example 1.1 below
indicates, the volumetric efficiency is useful for determination of the airflow rate of an
engine of a given displacement and speed.
EXAMPLE 1.1
Volumetric efficiency
A four-stroke 2.5 L direct injection automobile engine is tested on a dynamometer at a speed
of 2500 rpm. It produces a torque of 150 Nm, and its volumetric efficiency is measured to
be 0.85. What is the brake power 𝑊̇ b , and the mass airflow rate 𝑚̇ a through the engine?
The inlet air pressure and temperature are 75 kPa and 40◦ C.
SOLUTION The engine power 𝑊̇ b is
𝑊̇ b = 2𝜋𝜏𝑁 = (2𝜋)(150)(2500∕60) = 39.3 kW
The inlet air density is
𝜌i = 𝑃 ∕𝑅𝑇i = 75,000∕(287 × 313) = 0.835 kg∕m3
and the mass airflow rate 𝑚̇ a is
𝑚̇ a =
1
1
𝑒v 𝜌i 𝑉d 𝑁 = (0.85)(0.835)(2.5 × 10−3 )(2500∕60) = 3.70 × 10−2 kg∕s
2
2
Specific Fuel Consumption
The specific fuel consumption is a comparative metric for the efficiency of converting the
chemical energy of the fuel into work produced by the engine. As with the mean effective
pressure, there are two specific fuel consumption parameters, brake and indicated. The
brake specific fuel consumption (bsfc) is the fuel flow rate 𝑚̇ f , divided by the brake power
𝑊̇ b . It has three terms that are standard measurements in an engine test: the fuel flow rate,
the torque, and the engine speed:
bsf c =
𝑚̇ f
𝑚̇ f
=
̇
2𝜋𝜏𝑁
𝑊b
(1.18)
The indicated specific fuel consumption (isfc) is the ratio of the mass of fuel injected
during a cycle to the indicated cylinder work, and is used to compare engine performance
in computational simulations that do not include the engine friction.
𝑚
(1.19)
isf c = f
𝑊i
Typical values of measured bsfc for naturally aspirated automobile engines depend on the
engine load, with values ranging from about 200 to 400 g/kWh.
The specific fuel consumption and engine efficiency are inversely related, so that
the lower the specific fuel consumption, the greater the engine efficiency. Engineers use
Engine Performance Parameters
15
bsfc rather than thermal efficiency primarily because a more or less universally accepted
definition of thermal efficiency does not exist. We will explore the reasons why in Chapter 4.
Note for now only that there is an issue with assigning a value to the energy content of the
fuel. Let us call that energy the heat of combustion 𝑞c ; the brake thermal efficiency is then
𝜂=
𝑊̇ b
1
=
𝑚̇ f 𝑞c
bsf c 𝑞c
(1.20)
Inspection of Equation 1.20 shows that bsfc is a valid measure of efficiency provided 𝑞c is
held constant. Thus, two different engines can be compared on a bsfc basis provided that
they are operated with the same fuel.
EXAMPLE 1.2
Engine Parameters Calculation
A six-cylinder four-stroke automobile engine is being designed to produce 75 kW at 2000
rpm with a bsfc of 300 g/kWh and a bmep of 12 bar. The engine is to have equal bore
and stroke, and fueled with gasoline with a heat of combustion of 44,510 kJ/kg. (a) What
should be the design displacement volume and bore? (b) What is the mean piston speed at
the design point? (c) What is the fuel consumption per cycle per cylinder? (d) What is the
brake thermal efficiency?
SOLUTION (a) The displacement volume 𝑉d is
𝑉d =
𝑊̇ b
75
=
= 3.75 × 10−3 m3 = 3.75 L
bmep 𝑁∕2 (1200)(2000∕2)(1∕60)
𝑏=
(
𝑉d 4
𝑛c 𝜋
)1∕3
=
(
3.75 × 10−3 4
6
𝜋
)1∕3
= 92.7 mm
Most automobile engines have approximately a 90 mm bore and stroke.
(b) The mean piston speed is
𝑈̄ p = 2𝑁𝑠 = (2)(9.27 × 10−2 )(2000∕60) = 6.18 m∕s
(c) The cycle average fuel consumption rate per cylinder is
̄̇ f = bsf c × 𝑊̇ b ∕𝑛c = 300 × 75∕(6 × 60) = 62.5 g∕min
𝑚
so the mass of fuel injected per cylinder per cycle is
̄̇ f ∕(𝑁∕2) = 62.5∕(2000∕2) = 6.25 × 10−2 g
𝑚f = 𝑚
(d) The brake thermal efficiency is
𝜂=
1
3600
=
= 0.27
bsf c 𝑞c
(0.3)(44, 510)
16
Introduction to Internal Combustion Engines
Table 1.1 Performance Comparison of Three Different Four-Stroke Turbocharged Diesel
Engines
Parameter
# Cylinders
Bore (mm)
Stroke (mm)
Displacement per cylinder (L)
Power (kW)
Mass (kg)
Engine speed (rpm)
Mean piston speed (m/s)
Bmep (bar)
Power/volume (kW/L )
Mass/volume (kg/L )
Power/mass (kW/kg)
1.9 L
Automobile
5.9 L
Truck
7.2 L
Military
4
82
90
0.475
110
200
4000
12.05
17.3
57.9
105
0.55
6
102
120
0.983
242
522
3200
12.78
15.4
41.0
88
0.46
6
110
127
1.20
222
647
2400
10.16
15.4
30.8
90
0.35
Scaling of Engine Performance
The performance characteristics of three different diesel engines is compared in Table 1.1.
The engines are a four-cylinder 1.9 L automobile engine, a six-cylinder 5.9 L truck engine,
and a six-cylinder 7.2 L military engine. Comparison of the data in the Table indicates that
the performance characteristics of piston engines are remarkably similar when scaled to be
size independent. As Table 1.1 illustrates, the mean piston speed is about 12 m/s, the bmep
is about 15 bar, the power/volume is about 40 kW/L, and the power/mass about 0.5 kW/kg
for the three engines.
There is good reason for this; all engines tend to be made from similar materials. The
small differences noted could be attributed to different service criteria for which the engine
was designed. Since material stresses in an engine depend to a first order only on the bmep
and mean piston speed, it follows that for the same stress limit imposed by the material,
all engines should have the same bmep and mean piston speed. Finally, since the engines
geometrically resemble one another independent of size, the mass per unit displacement
volume is more or less independent of engine size.
1.5 ENGINE CONFIGURATIONS
Internal combustion engines can be built in many different configurations. For a given
engine, using a four- or two-stroke Otto or Diesel cycle, the configurations are characterized
by the piston--cylinder geometry, the inlet and exhaust valve geometry, the use of super or
turbochargers, the type of fuel delivery system, and the type of cooling system.
The reciprocating piston--cylinder combination remains the dominant form of the
internal combustion engine. Since the invention of the internal combustion engine, many
different piston--cylinder geometries have been designed, as shown in Figure 1.10. The
choice of a given arrangement depends on a number of factors and constraints, such as
engine balancing and available volume. The in-line engine is the most prevalent as it is
the simplest to manufacture and maintain. The V engine is formed from two in-line banks
of cylinders set at an angle to each other, forming the letter V. A horizontally opposed
or flat engine is a V engine with 180◦ offset piston banks. The W engine is formed from
Engine Configurations
(a) In line
17
(d) V
TDC
(b) Horizontally opposed
(c) Opposed piston
(crankshafts geared together)
(e) Radial
Figure 1.10 Various piston--cylinder geometries. Adapted from Obert (1950).
three in-line banks of cylinders set at an angle to each other, forming the letter W. A
radial engine has all of the cylinders in one plane with equal spacing between cylinder
axes. Radial engines are used in air-cooled aircraft applications, since each cylinder can
be cooled equally. Since the cylinders are in a plane, a master connecting rod is used
for one cylinder, and articulated rods are attached to the master rod. Alternatives to the
reciprocating piston--cylinder arrangement have also been developed, such as the rotary
Wankel engine, in which a triangular shaped rotor rotates eccentrically in a housing to
achieve compression, ignition, and expansion of a fuel--air mixture.
Engine Kinematics
Assuming a flat piston top, the instantaneous cylinder volume, 𝑉 (𝜃), at any crank angle is
𝑉 (𝜃) = 𝑉c +
𝜋 2
𝑏 𝑦
4
(1.21)
where y is the instantaneous stroke distance from top dead center:
By reference to Figure 1.6
𝑦 = 𝑙 + 𝑎 − [(𝑙2 − 𝑎2 sin2 𝜃)1∕2 + 𝑎 cos 𝜃 ]
(1.22)
The instantaneous volume 𝑉 (𝜃) can be nondimensionalized by the clearance volume
at top dead center, 𝑉tdc , resulting in
𝑦
𝑉 (𝜃)
= 1 + (𝑟 − 1)
𝑉̃ (𝜃) =
𝑉tdc
𝑠
(1.23)
We define a nondimensional parameter, 𝜖, the ratio of the crankshaft radius 𝑎 to the
connecting rod length 𝑙, as
𝜖=
𝑠
𝑎
=
𝑙
2𝑙
(1.24)
The value of 𝜖 for the slider--crank geometries used in modern engines is of order 1/3.
Therefore, the nondimensional piston displacement 𝑦∕𝑠 is
]
𝑦 1
1 [
1 − (1 − 𝜖 2 sin2 𝜃)1∕2
= (1 − cos 𝜃) +
𝑠
2
2𝜖
(1.25)
Introduction to Internal Combustion Engines
10
Dim. cylinder volume
18
Approx. volume
Exact volume
8
6
4
2
Figure 1.11 Cylinder
volume versus crank angle
for 𝑟 = 10, 𝜖 = 1∕3
(Equations 1.26 and 1.29).
0
−150
−100
−50
0
50
100
150
Crank angle (deg)
and the nondimensional cylinder volume 𝑉̃ (𝜃) is
]
(𝑟 − 1)
1 [
1 − (1 − 𝜖 2 sin2 𝜃)1∕2
(1 − cos 𝜃) +
𝑉̃ (𝜃) = 1 +
2
2𝜖
(1.26)
For 𝜖 < 1, we can expand the sin2 𝜃 term in a Taylor series,
(1 − 𝜖 2 sin2 𝜃)1∕2 ≃
1 2 2
𝜖 sin 𝜃 + 𝑂(𝜖 4 )
2
(1.27)
so
𝑦 1
𝜖
≃ (1 − cos 𝜃) + sin2 𝜃
𝑠
2
4
(1.28)
As 𝜖 → 0, the approximate volume 𝑉̃ (𝜃) can then be expressed as a function only of the
compression ratio 𝑟:
(𝑟 − 1)
(1 − cos 𝜃)
𝑉̃ (𝜃) ≃ 1 +
2
(1.29)
The cylinder volumes predicted by Equations 1.26 and 1.29 are compared in Figure 1.11
for a value of 𝜖 = 1∕3, using the MatlabⓇ program Volume.m listed in Appendix F.1.
Both equations give identical results at bottom dead center and top dead center, and since
the second term of the expansion is relatively small, the approximate volume relation
under-predicts the exact cylinder volume only by about 10% in the middle of the stroke.
The instantaneous piston velocity 𝑈p can be found by replacing 𝜃 with 𝜔𝑡 and differentiating Equation 1.25 with respect to time 𝑡 giving
[
]
𝑑𝑦 𝜔𝑠 sin(𝜔𝑡)
𝜖 cos 𝜔𝑡
𝑈p (𝜔𝑡) =
=
1+
(1.30)
𝑑𝑡
2
(1 − 𝜖 2 sin2 𝜔𝑡)1∕2
Equation 1.30 can be nondimensionalized by the mean piston speed 𝑈̄ p , resulting in
[
]
𝑈𝑝
𝜋
𝜖 cos 𝜃
= sin 𝜃 1 +
𝑈̃ p (𝜃) =
(1.31)
2
𝑈̄ 𝑝
(1 − 𝜖 2 sin2 𝜃)1∕2
Engine Configurations
19
1.8
Dim. piston velocity
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
Figure 1.12 Nondimensional
velocity versus crank angle for
𝜖 = 1∕3 (Equation 1.31).
0
0
50
100
Crank angle (deg)
150
Using the MatlabⓇ program Velocity.m listed in the Appendix F.2, the nondimensional
velocity 𝑈̃ 𝑝 (𝜃) is plotted versus crank angle from top dead center (tdc) to bottom dead center
(bdc) in Figure 1.12 for a value of 𝜖 = 1∕3. The piston velocity is zero at tdc and bdc. Due
to the geometry of the slider--crank mechanism, the velocity profile is nonsymmetric, with
the maximum nondimensional velocity of 𝑈̃ 𝑝 (𝜃) = 1.65 occurring at 72◦ atdc.
If we neglect terms of 𝑂(𝜖 2 ), and use the trigonometric identity sin2 𝜔𝑡 = (1 −
cos 2𝜔𝑡)∕2, the piston velocity can be approximated as
]
𝑑𝑦 𝜔𝑠 [
𝜖
sin 𝜔𝑡 + sin 2𝜔𝑡
≃
(1.32)
𝑈𝑝 =
𝑑𝑡
2
2
The acceleration 𝑎p is found by differentiating Equation 1.32 with respect to time
𝑑 2 𝑦 𝜔2 𝑠
≃
(1.33)
[cos 𝜔𝑡 + 𝜖 cos 2𝜔𝑡]
2
𝑑𝑡2
Note that the velocity and acceleration terms have two components, one varying with
the same frequency 𝜔 as the crankshaft, known as the primary term, and the other varying
at twice the crankshaft frequency 2𝜔, known as the secondary term. In the limit of an
infinitely long connecting rod, i.e., 𝜖 → 0, the motion reduces to a simple harmonic at a
frequency 𝜔.
The reciprocating motion of the connecting rod and piston creates accelerations and
thus inertial forces and moments that need to be considered in the choice of an engine configuration. In multicylinder engines, the cylinder arrangement and firing order are chosen
to minimize the primary and secondary forces and moments. Complete cancellation is possible for the following four-stroke engines: in-line 6- and 8-cylinder engines; horizontally
opposed 8- and 12-cylinder engines, and 12- and 16-cylinder V engines (Taylor, 1985).
𝑎p =
Intake and Exhaust Valve Arrangement
Gases are admitted and expelled from the cylinders by valves that open and close at the
proper times, or by ports that are uncovered or covered by the piston. There are many
design variations for the intake and exhaust valve type and location.
20
Introduction to Internal Combustion Engines
Rocker arm
Tappet
Tappet
clearance
Spring washer
Keeper
Outer spring
Inner
spring
Pushrod
Valve guide
Valve stem
Cam
follower
Lobe
Figure 1.13 Poppet valve
nomenclature (Taylor, 1985).
Valve-seat
insert
Valve seat
Valve head
Cam
Base circle
Poppet valves (see Figure 1.13) are the primary valve type used in internal combustion
engines, since they have excellent sealing characteristics. Sleeve valves have also been used,
but do not seal the combustion chamber as well as poppet valves. The poppet valves can
be located either in the engine block or in the cylinder head, depending on manufacturing
and cooling considerations. Older automobiles and small four-stroke engines have the
valves located in the block, a configuration termed underhead or L-head. Currently, most
engines use valves located in the cylinder head, an overhead or I-head configuration, as
this configuration has good inlet and exhaust flow characteristics.
The valve timing is controlled by a camshaft that rotates at half the engine speed for
four-stroke engine. A valve timing profile is shown in Figure 1.14. Lobes on the camshaft
along with lifters, pushrods, and rocker arms control the valve motion. Some engines use
an overhead camshaft to eliminate pushrods. The valve timing can be varied to increase
volumetric efficiency through the use of advanced camshafts that have moveable lobes, or
with electric valves. With a change in the load, the valve opening duration and timing can
be adjusted.
Superchargers and Turbochargers
All the engines discussed so far are naturally aspirated, i.e., as the intake gas is drawn in
by the downward motion of the piston. Engines can also be supercharged or turbocharged.
Supercharging is mechanical compression of the inlet air to a pressure higher than standard atmosphere by a compressor powered by the crankshaft. The compressor increases
the density of the intake air so that more fuel and air can be delivered to the cylinder to increase the power. The concept of turbocharging is illustrated in Figure 1.15.
Exhaust gas leaving an engine is further expanded through a turbine that drives a compressor. The benefits are twofold: (1) the engine is more efficient because energy that
21
Engine Configurations
TDC
0.9
TDC
POWER
BDC
EXHAUST
INTAKE
OVERLAP
BDC
COMPRESSION
TDC
0.8
0.7
Value lift (inches)
0.5
4
Exhaust value
opens
Intake value
opens
Exhaust value
closes
Intake value
closes
3
2
0.4
1
0.3
Piston location (inches)
5
0.6
0
Lobe separation
Spark plug fires 36°
before top dead center
ice. “timing is 36° advanced”
0.2
36°
0.1
Intake
Exhaust
62.3°
22.3°
59.9°
19.9°
0
0 20 40 60
BTDC
ATDC
Crank degrees
180 160 140 120 100 80 60 40 20 180 160140 120 100 80 60 40 20
BBDC
80 100 120 140 160 180 20 40 60 80 100 120 140 160180
ABDC
Figure 1.14 Poppet valve timing profile. (Courtesy of Competition Cams, Inc.)
Turbine
Compressor
Impeller
Exhaust
Turbine
wheel
Inlet
Exhaust
manifold
Intake
manifold
Figure 1.15 Turbocharger
schematic. (Courtesy of
Schwitzer.)
22
Introduction to Internal Combustion Engines
would have otherwise been wasted is recovered from the exhaust gas; and (2) a smaller
engine can be constructed to produce a given power because it is more efficient and because the density of the incoming charge is greater. The power available to drive the
compressor when turbocharging is a nonlinear function of engine speed such that at low
speeds there is little, if any, boost (density increase), whereas at high speeds the boost
is maximum. It is also low at part throttle and high at wide open throttle. These are desirable characteristics for an automotive engine since throttling or pumping losses are
minimized. Most large- and medium-sized diesel engines are turbocharged to increase their
efficiency.
Fuel Injectors and Carburetors
Revolutionary changes have taken place with computerized engine controls and fuel delivery systems in recent years and the progress continues. For example, the ignition and
fuel injection is computer controlled in engines designed for vehicular applications. Conventional carburetors in automobiles were replaced by throttle body fuel injectors in the
1980s, which in turn were replaced by port fuel injectors in the 1990s. Port fuel injectors
are located in the intake port of each cylinder just upstream of the intake valve, so there
is an injector for each cylinder. The port injector does not need to maintain a continuous
fuel spray, since the time lag for fuel delivery is much less than that of a throttle body
injector.
Direct injection spark ignition engines are available on many production engines. With
direct injection, the fuel is sprayed directly into the cylinder during the late stages of the
compression stroke. Compared with port injection, direct injection engines can be operated
at a higher compression ratio, and therefore will have a higher theoretical efficiency, since
they will not be knock limited. They will also be unthrottled, so they will have a greater
volumetric efficiency at part load. The evaporation of the injected fuel in the combustion
chamber will have a charge cooling effect, which will also increase its volumetric efficiency.
Cooling Systems
Some type of cooling system is required to remove the approximately 30% of the fuel
energy rejected as waste heat. Liquid and air cooling are the two main types of cooling
systems. The liquid cooling system (see Figure 1.16) is usually a single loop where a water
pump sends coolant to the engine block, and then to the head. Warm coolant flows through
the intake manifold to warm it and thereby assist in vaporizing the fuel. The coolant will
then flow to a radiator or heat exchanger, reject the waste heat to the atmosphere, and flow
Head
Radiator
Engine
block
Figure 1.16 Liquid
cooling system schematic.
Cylinder
heat
Heat
rejected
Water
pump
Examples of Internal Combustion Engines
23
Figure 1.17 Air cooling of model airplane
engine. (Courtesy R. Schroeder.)
back to the pump. When the engine is cold, a thermostat prevents coolant from returning
to the radiator, resulting in a more rapid warm-up of the engine. Liquid-cooled engines are
quieter than air-cooled engines, but have leaking, boiling, and freezing problems. Engines
with relatively low-power output, less than 20 kW, primarily use air cooling. Air cooling
systems use fins to lower the air side surface temperature (see Figure 1.17). There are
historical examples of combined water and air cooling. An early 1920s automobile, the
Mors, had a finned air-cooled cylinder and water-cooled heads.
1.6 EXAMPLES OF INTERNAL COMBUSTION ENGINES
Automotive Spark Ignition Four-Stroke Engine
A photograph of a V-6 3.2 L automobile engine is shown as in Figure 1.18 and in cutaway
view in Figure 1.19. The engine has a 89 mm bore and a stroke of 86 mm. The maximum
power is 165 kW (225 hp) at 5550 rpm. The engine has a single overhead camshaft per
Figure 1.18 3.2 L V-6 automobile
engine. (Courtesy of Honda
Motor Co.)
24
Introduction to Internal Combustion Engines
Figure 1.19 Cutaway view of 3.2 L
V-6 automobile engine. (Courtesy of
Honda Motor Co.)
piston bank with four valves per cylinder. The pistons are flat with notches for valve
clearance. The fuel is mixed with the inlet air by spraying the fuel into the intake port at
the Y-junction just above the intake valves.
As shown in Figure 1.20, the overhead camshaft acts on both the intake and exhaust
valves via rocker arms. The engine has variable valve timing applied to the intake valves
with a shift from low-lift short duration cam lobes to high-lift long duration cam lobes above
Variable
camshaft
Fuel
injector
Intake
Figure 1.20 A variable valve
timing mechanism. (Courtesy
of Honda Motor Co.)
Exhaust
Examples of Internal Combustion Engines
25
Figure 1.21 A 5.9 L L6 on-highway
diesel engine. (Courtesy of
PriceWebber.)
3500 rpm. In the low-lift short duration cam operation, the two intake valves have staggered
timing that creates additional swirl to increase flame propagation and combustion stability.
Roller bearings are used on the rocker arms to reduce friction. The clearance volume is
formed by an angled pent roof in the cylinder head, with the valves also angled.
Heavy Duty Truck Diesel Engine
A heavy duty truck diesel engine is shown in Figures 1.21. This engine is an inline sixcylinder turbocharged diesel engine with a 137-mm bore and 165-mm stroke for a total
displacement of 14.6 L. The rated engine power is 373 kW (500 hp). The compression ratio
is 16.5 to 1. The engine has electronically controlled, mechanically actuated fuel injectors,
and an overhead camshaft. Note that the cylinder head is flat, with the diesel fuel injector
mounted in the center of the combustion chamber. The inlet ports impart a swirl to the air
in the combustion chamber to improve mixing with the radial fuel spray.
The top of the piston has a Mexican hat-shaped crater bowl, so that the initial combustion will take place in the piston bowl. The injection nozzles have three to six holes through
which the fuel sprays into the piston bowl. The pressure required to spray the diesel fuel
into the combustion chamber is of the order of 1000 bar, for adequate spray penetration
into the bowl and subsequent atomization of the diesel fuel. The fuel injection pressure is
generated by a plunger driven by the camshaft rocker arm.
Stationary Gas Engine
A stationary natural gas engine is shown in Figures 1.22 and 1.23. Typical applications for
stationary engines include cogeneration, powering gas compressors, and power generation.
The engine shown in Figure 1.22 is an in-line eight-cylinder turbocharged engine, with
rated power of 1200 kW, bore of 240 mm, and stroke of 260 mm for a total displacement
of 94 L. The compression ratio is 10.9 to 1. This type of engine is designed to operate
at a constant speed condition, typically 1200 rpm. Each cylinder has two intake and two
exhaust valves. The piston has a combustion bowl with a deep dish concentrated near the
center of the piston, so most of the clearance volume is in the piston bowl.
Since natural gas engines are operated lean to reduce nitrogen oxides (NO𝑥 ), prechambers are used to initiate a stable combustion process. Pressurized natural gas is injected into
26
Introduction to Internal Combustion Engines
Figure 1.22 A 94 L L8 stationary
natural gas engine. (Courtesy of
Cooper Energy Services, Inc.)
Figure 1.23 Cutaway view of 94 L L8
stationary natural gas engine. (Courtesy
of Cooper Energy Services, Inc.)
a prechamber above the piston, and a spark plug in the prechamber is used to ignite the
natural gas. The increase in pressure projects the burning mixture into the main combustion
chamber, where the final stages of the combustion take place. Prechambers are also used
in high-speed diesel engines to achieve acceptable mixing and more complete combustion.
1.7 ALTERNATIVE POWER PLANTS
In this section, alternative power plants will be discussed in terms of a particular application
where they dominate the field by having some advantage over the internal combustion
engine.
Alternative Power Plants
27
First, consider electric motors which compete in the range of powers less than about
500 kW. They are used, for example, in forklifts operated within a factory or warehouse.
Internal combustion engines are not applied in this case because they would build up high
levels of pollutants such as carbon monoxide or nitric oxide. Electric motors are found in
a variety of applications, such as where the noise and vibration of a piston engine or the
handling of a fuel are unacceptable. Other examples are easy to think of in both indus trial
and residential sectors. Electric motors will run in the absence of air, such as in outer space
or under water; they are explosion proof; and they can operate at cryogenic temperatures. If
one can generalize, one might state with respect to electric motors that internal combustion
engines tend to be found in applications where mobility is a requirement or electricity is
not available.
Proponents of electric vehicles point out that almost any fuel can be used to generate
electricity, therefore we can reduce our dependence upon petroleum by switching to electric
vehicles. There would be no exhaust emissions emitted throughout an urban environment.
The emissions produced by the new electric generating stations could be localized geographically so as to minimize the effect. The main problem with electric vehicles is the
batteries used for energy storage. The electric vehicles that have been built to date have a
limited range of only 50--100 mi (80--160 km), on the order of one-fifth of what can be
easily realized with a gasoline engine powered vehicle. It is generally recognized that a
breakthrough in battery technology is required if electric vehicles are to become a significant part of the automotive fleet. Batteries have about 1% of the energy per unit mass of a
typical vehicular fuel, and a life span of about 5 years.
Hybrid electric vehicles (HEV), which incorporate a small internal combustion engine
with an electric motor and storage batteries, have been the subject of recent research, and
as of the year 2015, have reached the production stage, primarily due to their low fuel
consumption and emission levels. A hybrid electric vehicle has more promise than an
electric vehicle, since the HEV has an internal combustion engine to provide the energy to
meet vehicle range requirements. The battery then provides the additional power needed for
acceleration and climbing hills. The fuels used in the HEV engines in current production
include gasoline, diesel, and natural gas. Hybrid electric vehicles have a long history, as
the first HEV, the Woods Dual Power automobile, was introduced in 1916.
As shown in Figure 1.24, the engine and electric motor are placed in either a series
or parallel configuration. In a series configuration only the electric motor with power from
the battery or generator is used to drive the wheels. The internal combustion engine is
maintained at its most efficient and lowest emission operating points to run the generator
and charge the storage batteries. With the parallel configuration, the engine and electric
motor can be used separately or together to power the vehicle. The motors can be used as
generators during braking to increase vehicle efficiency.
The fuel cell electric vehicle (FCEV) is currently in the development phase, and
will be commercially available beginning in 2015. The chemical reaction in a fuel cell
produces lower emissions relative to combustion in an internal combustion engine. Recent
developments in proton exchange membrane (PEM) technology have been applied to
vehicular fuel cells. Current PEM fuel cells are small enough to fit beneath a vehicle’s
floor next to storage batteries and deliver 50 kW to an electric motor. The PEM fuel cell
requires a hydrogen fuel source to operate. Since there is presently no hydrogen fuel storage
infrastructure, on-board reforming of methanol fuel to hydrogen and CO2 is also required.
The reforming efficiency is about 60%, so coupled with a fuel cell efficiency of 70%, and
a motor efficiency of 90%, the overall fuel cell engine efficiency will be about 40%, about
the same as high-efficiency internal combustion engines.
Engine/
fuel conversion
unit
Batteries
Gen/
alt.
Inertial load
Viscous
load
Introduction to Internal Combustion Engines
Motor
Controller
(a) Series configuration
Clutch
Controller
Batteries
Figure 1.24 Hybrid electric
vehicle powertrain
configurations.
Engine/
fuel conversion
unit
Inertial load
Viscous
load
28
Motor
Clutch
(b) Parallel configuration
Gas turbine engines compete with internal combustion engines on the other end of
the power spectrum, at powers greater than about 500 kW. The advantages offered depend
on the application. Factors to consider are the efficiency and power per unit weight. A
gas turbine consists basically of a compressor--burner--turbine combination that provides a
supply of hot, high-pressure gas. This may then be expanded through a nozzle (turbojet),
through a turbine, to drive a fan, and then through a nozzle (turbofan), through a turbine, to
drive a propeller (turboprop), or through a turbine to spin a shaft in a stationary or vehicular
application.
One advantage a gas turbine engine offers to the designer is that the hardware responsible for compression, combustion, and expansion are three different devices, whereas in
a piston engine all these processes are done within the cylinder. The hardware for each
process in a gas turbine engine can then be optimized separately; whereas in a piston engine
compromises must be made with any given process, since the hardware is expected to do
three tasks. However, it should be pointed out that turbochargers give the designer of conventional internal combustion engines some new degrees of freedom toward optimization.
With temperature limits imposed by materials, the reciprocating engine can have a
greater peak cycle temperature than the gas turbine engine. In an internal combustion
engine, the gases at any position within the engine vary periodically from hot to cold. Thus,
the average temperature during the heat transfer to the walls is neither very hot nor cold.
On the other hand, the gas temperature at any position in the gas turbine is steady, and the
turbine inlet temperature is always very hot, thus tending to heat material at this point to a
greater temperature than anywhere in a piston engine.
The thermal efficiency of a gas turbine engine is highly dependent upon the adiabatic
efficiency of its components, which in turn is highly dependent upon their size and their
operating conditions. Large gas turbines tend to be more efficient than small gas turbines.
References
29
That airliners are larger than automobiles is one reason gas turbines have displaced piston
engines in airliners, but not in automobiles. Likewise gas turbines are beginning to penetrate
the marine industry, though not as rapidly, as power per unit weight is not as important
with ships as with airplanes.
Another factor favoring the use of gas turbines in airliners (and ships) is that the time
the engine spends operating at part or full load is small compared to the time the engine
spends cruising, therefore the engine can be optimized for maximum efficiency at cruise.
It is a minor concern that at part load or at take-off conditions the engine’s efficiency is
compromised. Automobiles, on the other hand, are operated over a wide range of load and
speed so a good efficiency at all conditions is better than a slightly better efficiency at the
most probable operating condition and a poorer efficiency at all the rest.
Steam- or vapor-cycle engines are much less efficient than internal combustion engines,
since their peak temperatures are about 800 K, much lower than the peak temperatures
(≈2500 K) of an internal combustion engine. They are used today almost totally in stationary
applications and where the energy source precludes the use of internal combustion engines.
Such energy sources include coal, waste feed stocks, nuclear, solar, and waste heat in the
exhaust gas of combustion devices including internal combustion engines.
In some applications, engine emission characteristics might be a controlling factor.
In the 1970s, in fact, a great deal of development work was done toward producing an
automotive steam engine when it was not known whether the emissions from the internal
combustion engine could be reduced enough to meet the standards dictated by concern for
public health. However, the development of catalytic converters, as discussed in Chapter 9,
made it possible for the internal combustion engine to meet emission standards at that time,
and remain a dominant prime mover technology.
The references of this introductory chapter contain a listing of both historical and
current books that will provide additional information about internal combustion engine
design, analysis, and performance. These books give the reader a deep appreciation of how
much the technology of internal combustion engines has advanced in the last century. In
chronological order, these books are Clerk (1910), Ricardo (1941), Benson and Whitehouse
(1979), Heywood (1988), Cummins (1989), Arcoumanis (1998), Stone (1999), Lumley
(1999), Pulkrabek (2003), Shi et al. (2011), Manning (2012), and Crolla et al. (2015).
1.8 REFERENCES
ARCOUMANIS, C. (1988), Internal Combustion Engines, Academic Press, London, England.
ARMSTRONG, D. and G. STIRRAT (1982), ‘‘Ford’s 1982 3.8L V6 Engine,’’ SAE paper 820112.
BENSON, R. and N. WHITEHOUSE (1979), Internal Combustion Engines, Pergamon Press, New York.
CLERK, D. (1910), The Gas, Petrol, and Oil Engine, Longmans, Green, and Co., London, England.
CROLLA, D., Ed. (2015), Encylopedia of Automotive Engineering, Wiley, New York.
CUMMINS, L. (1989), Internal Fire, Society of Automotive Engineers, Warrendale, Pennsylvania.
HEYWOOD, J. B. (1988), Internal Combustion Engine Fundamentals, McGraw-Hill, New York.
LUMLEY, J. (1999), Engines: An Introduction, Cambridge University Press, Cambridge, England.
MANNING, J. (2012), Internal Combustion Engine Design, Ricardo UK Limited, West Sussex,
England.
OBERT, E. (1950), Internal Combustion Engines, International Textbook Co., Scranton, Pennsylvania.
PULKRABEK, W. (2003), Engineering Fundamentals of the Internal Combustion Engine, Prentice Hall,
New York.
RICARDO, H. R. (1941), The High Speed Internal Combustion Engine, Interscience Publishers, New
York.
SHI, Y., H. GE, and R. REITZ (2011), Computational Optimization of Internal Combustion Engines,
Springer-Verlag, London, England.
30
Introduction to Internal Combustion Engines
Table 1.2 Engine Data for Homework Problems
Engine
Marine
Truck
Airplane
Bore
(mm)
Stroke
(mm)
Cylinders
Speed
(rpm)
Power
(kW)
136
108
86
127
95
57
12
8
8
2600
6400
10,500
1118
447
522
STONE, R. (1999), Introduction to Internal Combustion Engines, SAE International, Warrendale,
Pennsylvania.
TAYLOR, C. (1985), The Internal Combustion Engine in Theory and Practice, Vols. 1 and 2, MIT
Press, Cambridge, Massachusetts.
1.9 HOMEWORK
1.1
Compute the mean piston speed, bmep (bar), torque (Nm), and the power per piston area
for the engines listed in Table 1.2
1.2
A six-cylinder two-stroke engine with a compression ratio 𝑟 = 9 produces a torque of 1100
Nm at a speed of 2100 rpm. It has a bore 𝑏 of 123 mm and a stroke 𝑠 of 127 mm. (a) What
is the displacement volume and the clearance volume of a cylinder? (b) What is the engine
bmep, brake power, and mean piston speed?
1.3
A four-cylinder 2.5 L spark-ignited engine is mounted on a dyno and operated at a speed
of 𝑁 = 3000 rpm. The engine has a compression ratio of 10:1 and mass air--fuel ratio of
15:1. The inlet air manifold conditions are 80 kPa and 313 K. The engine produces a torque
of 160 Nm and has a volumetric efficiency of 0.82. (a) What is the brake power 𝑊̇ 𝑏 (kW)?
(b) What is the brake specific fuel consumption bsfc (g/kWh)?
1.4
The volumetric efficiency of the fuel injected marine engine in Table 1.2 is 0.80 and the
inlet manifold density is 50% greater than the standard atmospheric density of 𝜌𝑎𝑚𝑏 = 1.17
kg/m3 . If the engine speed is 2600 rpm, what is the air mass flow rate (kg/s)?
1.5
A 380 cc single-cylinder two-stroke motorcycle engine is operating at 5500 rpm. The
engine has a bore of 82 mm and a stroke of 72 mm. Performance testing gives a bmep =
6.81 bar, bsfc = 0.49 kg/kWh, and delivery ratio of 0.748. (a) What is the fuel to air ratio?
(b) What is the air mass flow rate (kg/s)?
1.6
A 3.8 L four-stroke four-cylinder fuel-injected automobile engine has a power output of 88
kW at 4000 rpm and volumetric efficiency of 0.85. The bsfc is 0.35 kg/kW h. If the fuel
has a heat of combustion of 42 MJ/kg, what are the bmep, thermal efficiency, and air to
fuel ratio? Assume atmospheric conditions of 298 K and 1 bar.
1.7
A 4.0 L six-cylinder engine is operating at 3000 rpm. The engine has a compression ratio
of 10:1, and volumetric efficiency of 0.85. If the bore and stroke are equal, what is the
stroke, the mean piston speed, cylinder clearance volume, and air mass flow rate into the
engine? Assume standard inlet conditions.
1.8
Chose an automotive, marine, or aviation engine of interest, and compute the engine’s
mean piston speed, bmep, power/volume, mass/volume, and power/mass. Compare your
calculated values with those presented in Table 1.1.
Homework
31
1.9
Compare the approximate, Equation 1.29, and exact, Equation 1.26, dimensionless cylinder
volume versus crank angle profiles for 𝑟 = 8, 𝑠 = 100 mm, and 𝑙 = 150 mm. What is the
maximum error and at what crank angle does it occur?
1.10
Plot the dimensionless piston velocity for an engine with a stroke 𝑠 = 100 mm and connecting rod length 𝑙 = 150 mm.
1.11
Assuming that the mean effective pressure, mean piston speed, power per unit piston area,
and mass per unit displacement volume are all size independent, how will the power per
unit weight of an engine depend upon the number of cylinders if the total displacement is
constant? To make the analysis easier, assume that the bore and stroke are equal.
Chapter
2
Heat Engine Cycles
2.1 INTRODUCTION
Studying heat engine cycles as simplified models of internal combustion engine processes
is very useful for illustrating the important parameters influencing engine performance.
Heat engine cycle analysis treats the combustion process as an equivalent heat addition
to an ideal gas. By modeling the combustion process as a heat addition, the analysis is
simplified since the details of the physics and chemistry of combustion are not required.
The various combustion processes are modeled as constant volume, constant pressure, or
finite energy release processes.
The internal combustion engine is not a heat engine, since it relies on internal combustion processes to produce work. However, heat engine models are useful for introducing the
idealized cycle parameters that are also used in more complex combustion cycle models,
for example, the fuel--air cycle, to be introduced in Chapter 4. The fuel--air cycle accounts
for the change in composition of the fuel--air mixture during the combustion process.
This chapter also provides a review of closed system and open system thermodynamics.
This chapter first uses a first law closed system analysis to model the compression and
expansion strokes and then incorporates open system control volume analysis of the intake
and exhaust strokes. An important parameter in the open system analysis is the residual
fraction of combustion gas, 𝑓 , remaining in the cylinder at the end of the exhaust stroke.
The scientific theory of heat engine cycles was first developed by Sadi Carnot (1796-1832), a French engineer, in 1824. His theory has two main axioms. The first axiom is that
in order to use a flow of energy to generate power, there must be two bodies at different temperatures, a hot body and a cold body. Work is extracted from the flow of energy from the hot
body to the cold body or reservoir. The second axiom is that there must be at no point a useless flow of energy, so heat transfer at a constant temperature is needed. Carnot developed
an ideal heat engine cycle, which is reversible; that is, if the balance of pressures is altered,
the cycle of operation is reversed. The efficiency of this cycle, known as the Carnot cycle, is
a function only of the reservoir temperatures, and the efficiency is increased as the temperature of the high-temperature reservoir is increased. The Carnot cycle, since it is reversible,
is the most efficient possible, and is the standard to which all real engines are compared.
Let us assume, to keep our mathematics simple, that the gas cycles analyzed in this
chapter use air with a constant specific heat as the working fluid. This assumption results
in simple analytical expressions for the efficiency as a function of the compression ratio.
A plot of the specific heat ratio, 𝛾 = 𝑐𝑝 /𝑐𝑣 , of air as a function of temperature is given in
Appendix A. Typical values of 𝛾 chosen for gas cycle calculations range from 1.3 to 1.4,
to correspond with measured cylinder temperature data.
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
32
Constant Volume Heat Addition
33
In performing a gas cycle computation, the heat addition 𝑄in (kJ) is required. It can be
estimated from the heat of combustion, 𝑞𝑐 (kJ/kgfuel ), of a fuel as follows:
𝑄in = 𝑚f 𝑞𝑐 = 𝑚𝑞in
(2.1)
where 𝑚f is the mass of fuel injected into the cylinder, 𝑚 is the mass of the fuel--air gas
mixture in the cylinder, and 𝑞in is the heat addition per unit mass of fuel--air mixture
(kJ/kgmix ). The mass of the fuel--air mixture can be determined using the ideal gas law
with known cylinder volume, mixture molecular mass, inlet pressure, and temperature, as
shown later in the text.
2.2 CONSTANT VOLUME HEAT ADDITION
This cycle is often referred to as the Otto cycle and considers the idealized case of an
internal combustion engine whose combustion is so rapid that the piston does not move
during the combustion process, and thus combustion is assumed to take place at constant
volume. The Otto cycle is named after Nikolaus Otto (1832--1891) who developed a fourstroke engine in 1876. Otto is considered the inventor of the modern internal combustion
engine, and founder of the internal combustion engine industry.
The Otto cycle engine is also called a spark ignition engine since a spark is needed to
initiate the combustion process. As we shall see, the combustion in a spark ignition engine
is not necessarily at constant volume. The working fluid in the Otto cycle is assumed to be
an ideal gas. The constant volume heat addition 𝑄in is non dimensionalized by the initial
pressure 𝑃1 and volume 𝑉1 . The Otto cycle example plotted in Figure 2.1 has a heat addition
𝑄in ∕𝑃1 𝑉1 = 20, a compression ratio 𝑟 = 8, and a specific heat ratio 𝛾 = 1.4.
The state processes for the Otto cycle are plotted in Figure 2.1. The four basic processes
are
1 to 2
2 to 3
3 to 4
4 to 1
isentropic compression
constant volume heat addition
isentropic expansion
constant volume heat rejection
The compression ratio of an engine is
𝑟=
𝑉1
𝑉2
(2.2)
The reader should be able to show that the following thermodynamic relations for the
Otto cycle processes are valid:
Compression stroke
𝑃2
= 𝑟𝛾
𝑃1
𝑇2
= 𝑟𝛾−1
𝑇1
(2.3)
Constant volume heat addition
𝑄in = 𝑚𝑐𝑣 (𝑇3 − 𝑇2 )
(2.4)
𝑇3
𝑄
= (𝛾 − 1) in 𝑟1−𝛾 + 1
𝑇2
𝑃1 𝑉1
(2.5)
𝑃3
𝑇
= 3
𝑃2
𝑇2
(2.6)
Heat Engine Cycles
25
20
Temperature (T/T1)
Internal energy
u – u1
P1V1
3
15
10
4
5
2
1
0
0
0.4
0.8
Entropy
1.2
1.6
s – s1
cv
100
imep/P1 = 12.9
80
3
Qin
P1V1
Pressure (P/P1)
34
= 20
60
40
20
2
4
1
Figure 2.1 The Otto
cycle (𝛾 = 1.40, 𝑟 = 8).
0
2
4
6
8
10
Volume (V/V2)
Expansion stroke
𝑃4 ( 1 )𝛾
=
𝑃3
𝑟
𝑇4 ( 1 )𝛾−1
=
𝑇3
𝑟
(2.7)
Heat rejection
𝑄out = 𝑚𝑐𝑣 (𝑇4 − 𝑇1 )
where
𝑚 = mass of gas in the cylinder, 𝑃1 𝑉1 ∕𝑅𝑇1
𝑐𝑣 = constant volume specific heat
𝑟 = compression ratio
𝛾 = specific heat ratio
(2.8)
35
Constant Volume Heat Addition
The thermal efficiency is given by the usual definition:
𝜂=
𝑄
𝑊out
= 1 − out
𝑄in
𝑄in
(2.9)
If we introduce the previously cited relations for 𝑄in , Equation 2.4, and 𝑄out , Equation 2.8, we get
𝜂 =1−
(𝑇4 − 𝑇1 )
1
=1−
𝛾−1
(𝑇3 − 𝑇2 )
𝑟
(2.10)
This cycle analysis indicates that the thermal efficiency 𝜂 of the Otto cycle depends
only on the specific heat ratio and the compression ratio. Figure 2.2 plots the thermal
efficiency versus compression ratio for a range of specific heat ratios from 1.2 to 1.4. The
efficiencies we have computed, for example, 𝜂 ∼ 0.56, for 𝑟 = 8 and 𝛾 = 1.40, are about
twice as large as those measured for actual engines. There are a number of reasons for this.
We have not accounted for internal friction and the combustion of a fuel within the engine,
and we have ignored heat transfer losses.
The indicated mean effective pressure (imep) is
𝑄
imep
𝑟
= 𝜂 in
𝑃1
𝑃1 𝑉1 𝑟 − 1
(2.11)
0.8
1.4
1.3
0.6
1.2
0.4
0.2
0
0
5
10
15
20
Specific heat ratio
Thermal efficiency
1.0
25
30
20
30
Qin
P1V1
40
25
20
Heat in
Indicated mean effective pressure
(imep/P1)
Compression ratio (r)
15
10
10
5
0
0
5
10
15
Compression ratio (r)
20
25
Figure 2.2 Otto cycle thermal efficiency and imep as a function of compression ratio and heat
addition.
36
Heat Engine Cycles
Note that the imep is nondimensionalized by the initial pressure 𝑃1 . The indicated
mean effective pressure is plotted versus compression ratio and heat addition in Figure 2.2
for 𝛾 = 1.30. As shown by Equation 2.11, the imep increases linearly with heat addition
and to a lesser degree with compression ratio.
Compression ratios found in actual spark ignition engines typically range from 9 to
11. The compression ratio is limited by two practical considerations: material strength and
engine knock. The maximum pressure, 𝑃3 , of the cycle scales with compression ratio as 𝑟𝛾 .
Engine heads and blocks have a design maximum stress, which should not be exceeded,
thus limiting the compression ratio. In addition, the maximum temperature 𝑇3 also scales
with the compression ratio as 𝑟𝛾 . If 𝑇3 exceeds the autoignition temperature of the air--fuel
mixture, combustion will occur ahead of the flame, a condition termed knock. The pressure
waves that are produced are damaging to the engine, and they reduce the combustion
efficiency. The knock phenomenon is discussed further in Chapter 7.
2.3 CONSTANT PRESSURE HEAT ADDITION
This cycle is often referred to as the Diesel cycle and models a heat engine cycle in which
energy is added at a constant pressure. The Diesel cycle is named after Rudolph Diesel
(1858--1913), who in 1897 developed an engine designed for the direct injection, mixing,
and autoignition of liquid fuel into the combustion chamber. The Diesel cycle engine is
also called a compression ignition engine. As we will see, actual diesel engines do not have
a constant pressure combustion process.
The cycle for analysis is shown in Figure 2.3. The four basic processes are
1 to 2
2 to 3
3 to 4
4 to 1
isentropic compression
constant pressure heat addition
isentropic expansion
constant volume heat rejection
Again assuming constant specific heats, the student should recognize the following
equations:
Heat addition
𝑄in = 𝑚𝑐𝑝 (𝑇3 − 𝑇2 )
(2.12)
( )𝛾
𝛽
𝑟
(2.13)
Expansion stroke
𝑃4
=
𝑃3
𝑇4
=
𝑇3
( )𝛾−1
𝛽
𝑟
where we have defined the parameter 𝛽, a measure of the combustion duration, as
𝛽=
𝑉3
𝑇
= 3
𝑉2
𝑇2
In this case, the indicated efficiency is
𝜂 =1−
1
𝑟𝛾−1
[
𝛽𝛾 − 1
𝛾(𝛽 − 1)
(2.14)
]
(2.15)
The term in brackets in Equation 2.15 is greater than 1, so that for the same compression
ratio 𝑟, the efficiency of the Diesel cycle is less than that of the Otto cycle. However, since
Diesel cycle engines are not knock limited, they operate at about twice the compression
Limited Pressure Cycle
8
3
20
7
6
15
5
10
4
4
5
2
0
1
3
2
Temperature (T/T1)
Internal energy
u – u1
P1V1
25
37
1
0
0.5
1.0
1.5
2.0
s–s
Entropy c 1
v
50
2
3
imep/P1 = 10.7
Pressure (P/P1)
40
Qin
P1V1
= 20
30
20
10
4
1
Figure 2.3 The Diesel
cycle (𝛾 = 1.30, 𝑟 = 20).
2
4
8
12
16
20
Volume (V/V2)
ratio of Otto cycle engines. For the same maximum pressure, the efficiency of the Diesel
cycle is greater than that of the Otto cycle. Diesel cycle efficiencies are shown in Figure 2.4
for a specific heat ratio of 1.30. They illustrate that high compression ratios are desirable
and that the efficiency decreases as the heat input increases. As 𝛽 approaches 1, the Diesel
cycle efficiency approaches the Otto cycle efficiency.
Although Equation 2.15 is correct, its utility suffers somewhat in that 𝛽 is not a natural
choice of independent variable. Rather, in engine operation, we think more in terms of the
heat transferred in. The two are related according to Equation 2.16.
𝛽 =1+
𝛾 − 1 𝑄in 1
𝛾 𝑃1 𝑉1 𝑟 𝛾−1
(2.16)
2.4 LIMITED PRESSURE CYCLE
Modern compression ignition engines resemble neither the constant volume nor the constant
pressure cycle, but rather a cycle in which some of the heat is added at constant volume and
Heat Engine Cycles
0.4
V3 = V4
0.2
0
Figure 2.4 Diesel cycle
characteristics as a
function of compression
ratio and heat addition
(𝛾 = 1.30).
0
20
40
Otto cycle
5
10
15
Compression ratio (r)
20
Heat in
0.6
Q
P1V1
0.8
25
30
20
40
15
30
10
20
10
5
0
0
5
10
15
Compression ratio (r)
20
Q
P1V1
25
Heat in
Thermal efficiency
1.0
Indicated mean effective pressure
(imep/P1)
38
0
25
then the remaining heat is added at constant pressure. This limited pressure or ‘‘dual’’ cycle
is a gas cycle model that can be used to model combustion processes that are slower than
constant volume, but more rapid than constant pressure. The limited pressure cycle can also
provide algebraic equations for performance parameters such as the thermal efficiency and
imep. The distribution of heat added in the two processes is something an engine designer
can specify approximately by choice of fuel, the fuel injection system, and the engine
geometry to limit the peak pressure in the cycle.
The cycle notation is illustrated in Figure 2.5. In this case, we have the following
equation 2.17 for 𝑄in :
Heat addition
𝑄in = 𝑚𝑐𝑣 (𝑇2.5 − 𝑇2 ) + 𝑚𝑐𝑝 (𝑇3 − 𝑇2.5 )
(2.17)
The expansion stroke is still described by Equation 2.14 provided we write 𝛽 = 𝑉3 ∕𝑉2.5 .
If we let 𝛼 = 𝑃3 ∕𝑃2 , a pressure rise parameter, it can be shown that
𝜂 =1−
1
𝑟𝛾−1
𝛼𝛽 𝛾 − 1
𝛼 − 1 + 𝛼𝛾(𝛽 − 1)
(2.18)
The constant volume and constant pressure cycles can be considered as special cases
of the limited pressure cycle in which 𝛽 = 1 and 𝛼 = 1, respectively. The use of the limited
pressure cycle model requires information about either the fractions of constant volume
and constant pressure heat addition or the maximum pressure, 𝑃3 . A common assumption
is to equally split the heat addition. Results for the case of 𝑃3 /𝑃1 = 50 and 𝛾 = 1.3 are
shown in Figure 2.6, showing efficiencies and imep that are between the Otto and Diesel
limits. For the same compression ratio, the Otto cycle has the largest net work, followed
Miller Cycle
39
30
Internal energy
u – u1
P1V1
3
25
20
Isometric
15
10
4
Isobaric
2.5
5
2
0
1
0
2.5
50
0.5
1.0
1.5
s – s1
Entropy
cv
2.0
2.5
3.0
3
imep/P1 = 15.3
Pressure (P/P1)
40
Qin
P1V1
= 30
2
30
20
10
4
0
5
10
1
15
Volume (V/V2)
Figure 2.5 The limited pressure cycle (𝛾 = 1.30, 𝑟 = 15).
by the limited pressure and the Diesel. Transformation of 𝛽 and 𝛼 to more useful variables
yields
[
]
𝛾 − 1 𝑄in 1
𝛼−1
−
(2.19)
𝛽 =1+
𝛼𝛾
𝑃1 𝑉1 𝑟 𝛾−1 𝛾 − 1
𝛼=
1 𝑃3
𝑟𝛾 𝑃 1
(2.20)
2.5 MILLER CYCLE
The efficiency of an internal combustion engine will increase if the expansion ratio is
larger than the compression ratio. There have been many mechanisms of varying degrees
of complexity designed to produce different compression and expansion ratios, and thus
Heat Engine Cycles
Thermal efficiency
10
Figure 2.6 Comparison of
limited pressure cycle with
Otto and Diesel cycles
(𝛾 = 1.30).
Qin
= 30
P1V1
0.8
P3 /P1 = 50 for the dual cycle
0.6
0.4
Dual
Otto
0.2
0
Indicated mean effective pressure (imep/P1)
40
Diesel
0
5
10
15
Compression ratio (r)
20
25
20
Dual
15
Otto
Diesel
10
Qin
= 30
P1V1
5
P3 /P1 = 50 for dual cycle
0
0
5
10
15
Compression ratio (r)
20
25
greater efficiency. The Miller cycle (Miller, 1947), was patented by R. H. Miller (1890-1967), an American inventor, in 1957. It is a cycle that uses early or late inlet valve closing
to decrease the effective compression ratio. This cycle has been used in ship diesel engines
since the 1960s, and in the 1990s adopted by Mazda for use in vehicles. For example, a
2.3 L supercharged V6 Miller cycle engine was used as the replacement for a 3.3 L naturally
aspirated V6 engine in the 1995 Mazda Millenia. This engine used late inlet valve closing
at 30◦ after the start of the compression stroke.
A related cycle, the Atkinson cycle, is one in which the expansion stroke continues
until the cylinder pressure at point 4 decreases to atmospheric pressure. This cycle is named
after James Atkinson (1846--1914), an English engineer, who invented and built an engine
he named the ‘‘cycle’’ engine in 1889. This engine had a two-bar linkage between the
connecting rod and the crankshaft so that the piston traveled through four unequal strokes
in every crankshaft revolution. The expansion to intake stroke ratio was 1.78:1.
The Miller gas cycle is shown in Figure 2.7. In this cycle, as the piston moves downward
on the intake stroke, the cylinder pressure follows the constant pressure line from point 6
to point 1. For early inlet valve closing, the inlet valve is closed at point 1 and the cylinder
pressure decreases during the expansion to point 7. As the piston moves upward on the
compression stroke, the cylinder pressure retraces the path from point 7 through point 1
to point 2. The net work done along the two paths 1-7 and 7-1 cancel, so the effective
compression ratio 𝑟𝑐 = 𝑉1 / 𝑉2 is less than the expansion ratio 𝑟e = 𝑉4 / 𝑉3 .
41
Finite Energy Release
P
3
2
4
Pi , Pe
6
1
5
7
V
Figure 2.7 The Miller cycle.
For late inlet valve closing, a portion of the intake air is pushed back into the intake
manifold before the intake valve closes at point 1. Once the inlet valve closes, there is less
mixture to compress in the cylinder, and thus less compression work.
Performing a first law analysis of the Miller cycle, we first define the parameter 𝜆, the
ratio of the expansion ratio to the compression ratio:
𝜆 = 𝑟e ∕𝑟c
(2.21)
The heat rejection has two components:
𝑄out = 𝑚𝑐𝑣 (𝑇4 − 𝑇5 ) + 𝑚𝑐𝑝 (𝑇5 − 𝑇1 )
(2.22)
In this case, the thermal efficiency is
𝜂 = 1 − (𝜆𝑟c )1−𝛾 −
𝜆1−𝛾 − 𝜆(1 − 𝛾) − 𝛾 𝑃1 𝑉1
𝛾 −1
𝑄in
(2.23)
Equation 2.23 reduces to the Otto cycle thermal efficiency as 𝜆 → 1. The imep is:
𝑟c
𝑄
imep
= 𝜂 in
𝑃1
𝑃1 𝑉1 𝜆𝑟c − 1
(2.24)
The thermal efficiency of the Miller cycle is not only a function of the compression
ratio and specific heat ratio, but also a function of the expansion ratio and the load 𝑄in .
The ratio of the Miller cycle thermal efficiency to an equivalent Otto cycle efficiency with
the same compression ratio is plotted in Figure 2.8 for a range of compression ratios and 𝜆
values. For example, with 𝜆 = 2 and 𝑟c = 12, the Miller cycle is about 20% more efficient
than the Otto cycle. The ratio of the Miller/Otto cycle imep is plotted as a function of 𝜆
in Figure 2.9. As 𝜆 increases, the imep decreases significantly, since the fraction of the
displacement volume 𝑉d that is filled with the inlet fuel--air mixture decreases. This relative
decrease in imep and engine power is a disadvantage of the Miller cycle, which is the reason
supercharging of the inlet mixture is used to increase the imep.
2.6 FINITE ENERGY RELEASE
Energy Release Fraction
In the Otto and Diesel cycles, the fuel is assumed to burn at rates that result in constant
volume top dead center combustion or constant pressure combustion, respectively. Actual
Heat Engine Cycles
Efficiency ratio
1.4
1.3
rc = 8
10
1.2
12
Qin
= 30
P1V1
1.1
Figure 2.8 Ratio of
Miller to Otto cycle
thermal efficiency with
same compression ratio,
𝑟c (𝛾 = 1.30).
1
1
1.5
2
2.5
3
λ (re /rc)
1
Qin
= 30
P1V1
0.9
0.8
Miller/Otto imep ratio
42
0.7
0.6
rc
8
10
0.5
0.4
Figure 2.9 Ratio of
Miller to Otto cycle
imep with same
compression ratio,
𝑟c (𝛾 = 1.30).
12
0.3
0.2
1.00
1.50
2.00
2.50
3.00
λ (re /rc)
engine pressure and temperature profile data do not match these simple models, and more
realistic modeling, such as a finite energy release model, is required. A finite energy
release model is a differential equation model of an engine cycle in which the heat addition
is specified as a function of the crank angle. It is also known as a ‘‘zero-dimensional’’
model, since it is a function only of crank angle, and not a function of the combustion
chamber geometry.
Energy release models can address questions that the simple gas cycle models cannot.
For example, if one wants to know about the effect of spark timing or heat and mass transfer
on engine work and efficiency, an energy release model is required. Also if heat transfer is
included, as is done in Chapter 11, then the state changes for the compression and expansion
processes are no longer isentropic, and cannot be expressed as simple algebraic equations.
For further information about energy release models, also known as zero-dimensional
thermodynamic models since there is no engine spatial information used, the reader is
referred to Foster (1985).
Finite Energy Release
43
1
xb
θd
0
Figure 2.10 Cumulative energy release
function.
θs
Crank angle θ
A typical cumulative mass fraction burned, that is, fraction of fuel energy released,
curve for a spark ignition engine is shown in Figure 2.10. The figure plots the cumulative
mass fraction burned 𝑥b (𝜃) versus the crank angle. The characteristic features of the mass
fraction burned curve are an initial small slope region beginning with spark ignition and
the start of energy release at 𝜃s , followed by a region of rapid growth and then a more
gradual decay. The three regions correspond to the initial ignition development, a rapid
burning region, and a burning completion region. This S-shaped curve can be represented
analytically by a trigonometric function as indicated by Equation 2.25:
[
(
)]
𝜋(𝜃 − 𝜃s )
1
1 − cos
(2.25)
𝑥b (𝜃) =
2
𝜃d
or an exponential relation, known as a Wiebe function, as given in Equation 2.26:
[ (
) ]
𝜃 − 𝜃s 𝑛
(2.26)
𝑥b (𝜃) = 1 − exp −𝑎
𝜃d
where
𝑥b = fraction of energy release
𝜃 = crank angle
𝜃s = start of energy release
𝜃d = duration of energy release
𝑛 = Wiebe form factor
𝑎 = Wiebe efficiency factor
The Wiebe function is named after Ivan Wiebe (1902--1969), a Russian engineer who
developed an energy release model based on analysis of combustion chain reaction events
(Ghojel, 2010). The Wiebe function can be used for modeling the energy release in a
wide variety of combustion systems. For example, diesel engine combustion, which has a
premixed phase and a diffusion phase, can be modeled using a combined double Wiebe
function. The energy release curve for the diesel engine is double peaked due to the two
combustion phases, and discussed in more detail later in the diesel combustion section of
Chapter 7.
44
Heat Engine Cycles
Since the cumulative energy release curve asymptotically approaches a value of 1,
the end of combustion needs to defined by an arbitrary limit, such as 90, 99, or 99.9%
complete combustion; that is, 𝑥b = 0.90, 0.99, or 0.999, respectively. Corresponding values
of the Wiebe efficiency factor 𝑎 are 2.302, 4.605, and 6.908 respectively. The value of the
efficiency factor 𝑎 = 6.908 was used by Wiebe in his engine modeling calculations.
The values of the form factor 𝑛 and burn duration 𝜃d depend on the particular type
of engine, and to some degree on the engine load and speed. These parameters can be
deduced using experimental burn rate data, which in turn are obtained from the cylinder
pressure profile as a function of crank angle, discussed in more detail in the combustion
analysis section of Chapter 12. Values of 𝑎 = 5 and 𝑛 = 3 have been reported to fit well
with experimental data (Heywood, 1988).
The rate of energy release for the Wiebe function as a function of crank angle, Equation 2.27, is obtained by differentiation of the cumulative energy release function.
𝑑𝑥
𝑑𝑄
= 𝑄in b
𝑑𝜃
𝑑𝜃
= 𝑛𝑎
𝑄in
(1 − 𝑥b )
𝜃d
(
𝜃 − 𝜃s
𝜃d
)𝑛−1
(2.27)
The computer program BurnFraction.m is listed in Appendix F, and can be used
to plot the Wiebe function cumulative and rate of energy release for different engine
conditions. The use of the program is detailed in the following example.
EXAMPLE 2.1
Energy Release Fractions
Using the Wiebe function, plot the cumulative and the rate of energy release for a combustion event with the start of energy release at 𝜃s = −20◦ and the duration of energy release
𝜃d = 60◦ . Assume the Wiebe efficiency factor 𝑎 = 5, that is, 𝑥b = 0.9933, and the Wiebe
form factor 𝑛 = 4.
SOLUTION The above parameters are entered into the computer program BurnFraction.m
as shown below, and the resulting plots are shown in Figures 2.11 and 2.12.
Comment: Note the asymmetry of the burn rate, as a result of the form factor value, and the
peak value of the burn rate at 18◦ atdc. As discussed in more detail in the next example,
optimal work from an engine usually occurs with a peak burn rate a few degrees after
top dead center, so a significant fraction of the combustion will be occurring during the
expansion process.
function [ ]=BurnFraction( )
This program computes and plots the cumulative burn fraction
and the instantaneous burn rate
a = 5;
Wiebe efficiency factor
n = 4;
Wiebe form factor
thetas = -20;
start of combustion
thetad = 60;
duration of combustion
....
Finite Energy Release
45
Cumulative burn fraction
1
0.8
0.6
0.4
0.2
0
20
Figure 2.11 Cumulative energy
release curve for Example 2.1.
10
0
10
20
30
40
30
40
Crank angle θ (deg)
2.5
Burn rate (J/deg)
2
1.5
1
0.5
Figure 2.12 Rate of energy
release curve for Example 2.1.
0
20
10
0
10
20
Crank angle θ (deg)
Energy Equation
We now develop a simple finite energy release model by incorporating the Wiebe function
equation, Equation 2.27, into the differential energy equation. We assume that the energy
release occurs for a given combustion duration 𝜃d during the compression and expansion
strokes, and solve for the resulting cylinder pressure 𝑃 (𝜃) as a function of crank angle. The
simple model assumes the inlet and exhaust valves are closed at the start of integration at
𝜃 = −180◦ , so it does not account for flow into and out of the combustion chamber.
As shown in the following derivation, the differential form of the energy equation does
not have a simple analytical solution due to the finite energy release term. It is integrated
numerically, starting at bottom dead center, compressing to top dead center, and then
expanding back to bottom dead center.
The closed system differential energy equation (note that work and heat interaction
terms are not true differentials) for a small crank angle change, 𝑑𝜃, is
𝛿𝑄 − 𝛿𝑊 = 𝑑𝑈
(2.28)
46
Heat Engine Cycles
since 𝛿𝑊 = 𝑃 𝑑𝑉 and 𝑑𝑈 = 𝑚𝑐𝑣 𝑑𝑇 ,
𝛿𝑄 − 𝑃 𝑑𝑉 = 𝑚𝑐𝑣 𝑑𝑇
(2.29)
Assuming ideal gas behavior,
(2.30)
𝑃 𝑉 = 𝑚𝑅𝑇
which in differential form is
𝑚 𝑑𝑇 =
1
(𝑃 𝑑𝑉 + 𝑉 𝑑𝑃 )
𝑅
(2.31)
The energy equation is therefore
𝑐𝑣
(𝑃 𝑑𝑉 + 𝑉 𝑑𝑃 )
𝑅
differentiating with respect to crank angle, and introducing 𝑑𝑄 = 𝑄in 𝑑𝑥,
)
𝑐 ( 𝑑𝑉
𝑑𝑥
𝑑𝑉
𝑑𝑃
−𝑃
= 𝑣 𝑃
+𝑉
𝑄in
𝑑𝜃
𝑑𝜃
𝑅
𝑑𝜃
𝑑𝜃
Solving for the pressure 𝑃 ,
𝛿𝑄 − 𝑃 𝑑𝑉 =
(2.32)
(2.33)
𝑄 𝑑𝑥
𝑑𝑃
𝑃 𝑑𝑉
= −𝛾
+ (𝛾 − 1) in
(2.34)
𝑑𝜃
𝑉 𝑑𝜃
𝑉 𝑑𝜃
In practice, it is convenient to normalize the equation with the pressure 𝑃1 and volume
𝑉1 at bottom dead center:
𝑃̃ = 𝑃 ∕𝑃1
𝑉̃ = 𝑉 ∕𝑉1
𝑄̃ = 𝑄in ∕𝑃1 𝑉1
in which case we obtain
𝑄̃ 𝑑𝑥
𝑑 𝑃̃
𝑃̃ 𝑑 𝑉̃
= −𝛾
+ (𝛾 − 1)
̃
𝑑𝜃
𝑉 𝑑𝜃
𝑉̃ 𝑑𝜃
(2.35)
(2.36)
The differential equation for the work is
̃
𝑑 𝑉̃
𝑑𝑊
= 𝑃̃
𝑑𝜃
𝑑𝜃
(2.37)
In order to integrate Equations 2.36 and 2.37, an equation for the cylinder volume 𝑉̃ as
a function of crank angle is needed. By reference to Chapter 1, the dimensionless cylinder
volume 𝑉̃ (𝜃) = 𝑉 (𝜃)∕𝑉bdc = 𝑉 (𝜃)∕𝑉1 for 𝑙 >> 𝑠 is
𝑟−1
(1 − cos 𝜃)
𝑉̃ (𝜃) = 1 +
2𝑟
which upon differentiation gives
(2.38)
𝑟−1
𝑑 𝑉̃
=
sin 𝜃
(2.39)
𝑑𝜃
2𝑟
Equations 2.36 and 2.37 are linear first-order differential equations of the form
𝑑 𝑌̃ ∕𝑑𝜃 = 𝑓 (𝜃, 𝑌̃ ), and easily solved by numerical integration. Solution yields 𝑃̃ (𝜃) and
̃ (𝜃), which once determined allow computation of the net work of the cycle, the thermal
𝑊
efficiency, and the indicated mean effective pressure. Note that in this analysis we have
neglected heat and mass transfer losses, and will consider them in the next section.
The thermal efficiency is computed directly from its definition
𝜂=
̃
𝑊
𝑄̃
(2.40)
Finite Energy Release
The imep is then computed using Equation 2.41
)
(
imep
𝑟
= 𝜂 𝑄̃
𝑃1
𝑟−1
47
(2.41)
For the portions of the compression and expansion strokes before ignition and after
combustion, that is, where 𝜃 < 𝜃s and 𝜃 > 𝜃s + 𝜃d , the energy release term 𝑑𝑄∕𝑑𝜃 = 0,
allowing straightforward integration of the energy equation and recovery of the isentropic
pressure--volume relation:
𝑃̃ 𝑑 𝑉̃
𝑑 𝑃̃
= −𝛾
𝑑𝜃
𝑉̃ 𝑑𝜃
𝑑 𝑃̃
𝑑 𝑉̃
= −𝛾
̃
𝑃
𝑉̃
𝑃̃ 𝑉̃ 𝛾 = constant
(2.42)
(2.43)
(2.44)
The differential energy equation, Equation 2.34, can also be used in reverse to compute
energy release curves from experimental measurements of the cylinder pressure. This
procedure is discussed in detail in Chapter 12. Commercial combustion analysis software
is available to perform such analysis in real time during an experiment.
The computer program FiniteHeatRelease.m is listed in Appendix F, and can
be used to compare the performance of two different engines with different combustion
and geometric parameters. The program computes gas cycle performance by numerically
integrating Equation 2.34 for the pressure as a function of crank angle. The integration
starts at bottom dead center (𝜃 = −180◦ ), with initial inlet conditions 𝑃1 , 𝑉1 , 𝑇1 , the gas
molecular weight 𝑀, and specific heat ratio 𝛾 given. The integration proceeds degree by
degree to top dead center (𝜃 = 0◦ ) and back to bottom dead center. Once the pressure is
computed as a function of crank angle, the net work, thermal efficiency, and imep are also
computed. The use of the program is detailed in the following example.
EXAMPLE 2.2
Finite Energy Release
A single-cylinder spark ignition cycle engine is operated at full throttle, and its performance
is to be predicted using a Wiebe energy release analysis. The engine has a compression ratio
of 10. The initial cylinder pressure, 𝑃1 , at bottom dead center is 1 bar, with a temperature
𝑇1 at bottom dead center of 300 K. The bore and stroke of the engine are 𝑏 = 100 mm
and 𝑠 = 100 mm. The total heat addition 𝑄in = 1764 J and the combustion duration 𝜃d is
constant at 40◦ . Assume that the ideal gas specific heat ratio 𝛾 is 1.4, the molecular mass
of the gas mixture is 29 kg/kmol, and the Wiebe energy release parameters are 𝑎 = 5 and
𝑛 = 3.
(a) Compute the displacement volume 𝑉d , the volume at bottom dead center 𝑉1 , the dĩ and the mass of gas in the cylinder 𝑚.
mensionless heat addition 𝑄,
(b) Plot the pressure and temperature profiles versus crank angle for 𝜃s1 = −20◦ (engine 1)
and 𝜃s2 = 0◦ (engine 2).
(c) Determine the effect of changing the start of energy release from 𝜃s = −50◦ to 𝜃s =
+20◦ atdc on the thermal efficiency, and imep of the engine.
SOLUTION (a) The displacement volume is
𝑉d =
𝜋 2
𝑏 𝑠 = 7.85 × 10−4 m3
4
48
Heat Engine Cycles
The volume at bottom dead center is
𝑉1 =
𝑉d
7.85 × 10−4
=
= 8.73 × 10−4 m3
1 − 1∕𝑟
1 − 1∕10
The dimensionless heat addition is
𝑄̃ = 𝑄in ∕𝑃1 𝑉1 = 1764∕[(101 × 103 )(8.73 × 10−4 )] = 20
The mass of gas in the cylinder is
𝑚=
𝑃 1 𝑉1
(101)(8.73 × 10−4 )
=
= 1.03 × 10−3 kg
𝑅𝑇1
(8.314∕29)(300)
(b) The above engine parameters are entered into the FiniteHeatRelease.m
program as shown below. The start of energy release is 𝜃s = −20◦ for engine 1 and 𝜃s = 0◦
for engine 2, and all other parameters are the same for both engines.
function [ ] = FiniteHeatRelease( )
Gas cycle heat release code for two engines
Engine input parameters:
thetas(1,1) = -20;
Engine 1 start of heat release (deg)
thetas(2,1) = 0;
Engine 2 start of heat release (deg)
thetad(1,1) = 40;
Engine 1 duration of heat release (deg)
thetad(2,1) = 40;
Engine 2 duration of heat release (deg)
r =10;
Compression ratio
gamma = 1.4;
Ideal gas const
Q = 20.4;
Dimensionless total heat addition
a = 5;
Wiebe efficiency factor a
n = 3;
Wiebe exponent n
...}
The pressure profiles are compared in Figure 2.13. The pressure rise for engine 1 is
more than double that of engine 2. The maximum pressure of about 8800 kPa occurs at 11◦
after top dead center for engine 1, and at about 25◦ after top dead center for engine 2. The
temperature profiles are shown in Figure 2.14. Engine 1 has a peak temperature of about
2900 K, almost 400 K above that of engine 2.
(c) The start of heat release is varied from 𝜃s = −50◦ to 𝜃s = 0◦ , as shown in Figures 2.15 and 2.16, and the resulting thermal efficiency 𝜂 and imep are plotted.
Comment: The results indicate that there is an optimum crank angle for the start of energy
release that will maximize the thermal efficiency and imep. For this computation, the
optimum start of energy release is about 𝜃s = −20◦ , resulting in a maximum thermal
efficiency of about 60% and imep/𝑃1 of about 13.2. At crank angles less than or greater
than this optimal angle, the thermal efficiency and imep/𝑃1 decrease.
An explanation for the optimal crank angle is as follows. If the energy release begins too
early during the compression stroke, the negative compression work will increase, since
the piston is doing work against the increasing combustion gas pressure. Conversely, if
the energy release begins too late, the energy release will occur in an increasing cylinder
Finite Energy Release
49
8800
Engine 1
Engine 2
4400
P (kPa)
Figure 2.13 Pressure
profiles for Example 2.2.
–180
–90
0
Crank angle (deg)
90
180
3000
Engine 1
Engine 2
1500
T (K)
Figure 2.14 Temperature
profiles for Example 2.2.
–180
–90
0
Crank angle (deg)
90
180
volume, resulting in lower combustion pressure, and lower net work. In practice, the
optimum spark timing also depends on the engine load, and is in the range of 𝜃s = −30◦ to
𝜃s = −5◦ . The resulting location of the peak combustion pressure is typically between 5◦
and 15◦ atdc.
Cylinder Heat and Mass Transfer Loss
In this section, we develop simple models of the heat transfer and the mass blowby process,
and include them in the energy release analysis developed in the previous section. Engines
are air or water cooled to keep the engine block temperatures within safe operating limits,
so there is a significant amount of heat transfer from the combustion gas to the surrounding
cylinder walls. Also, internal combustion engines do not operate on closed thermodynamic
cycles, rather there is an induction of fresh charge and expulsion of combustion products,
and there is leakage of combustion gases or blowby past the rings, since the rings do not
provide a complete seal of the combustion chamber. The blowby can affect the indicated
performance, the friction and wear, and the hydrocarbon emissions of the engine.
The heat transfer to the cylinder walls is represented by a Newtonian-type convection
equation with a constant heat transfer coefficient h. More realistic models accounting for
a variable h are presented in Chapter 11. The mass flow is assumed to be blowby past
the rings from the combustion chamber at a rate proportional to the mass of the cylinder
contents. A useful rule of thumb is that new engines will have a 0.5% blowby, then operate
for most of their life at a typical level of 1% blowby, and gradually reach a maximum
blowby of 2.5--3.0% at the end of their useful life.
50
Heat Engine Cycles
The heat transfer to the walls can be included by expanding the energy release 𝑑𝑄 term
in the energy equation to include both heat addition and loss, as indicated in Equation 2.45:
𝑑𝑄 = 𝑄in 𝑑𝑥 − 𝑑𝑄l
(2.45)
𝑑𝑄l
= 𝒉𝐴(𝑇 − 𝑇w )
𝑑𝑡
(2.46)
The heat loss 𝑑𝑄l is
where
𝒉 = heat transfer coefficient
𝐴 = cylinder surface area in contact with the gases
𝑇w = cylinder wall temperature
The combustion chamber area 𝐴 is a function of crank angle 𝜃, and is the sum of the
combustion chamber area at top dead center 𝐴o and the instantaneous cylinder wall area
𝐴w (𝜃). The instantaneous combustion chamber area and volume are
𝐴 = 𝐴o + 𝜋 𝑏 𝑦(𝜃)
𝑉 = 𝑉o +
𝜋𝑏2
𝑦(𝜃)
4
or
𝐴 = (𝐴o − 4𝑉o ∕𝑏) + 4𝑉 ∕𝑏
(2.47)
where 𝑉o is the cylinder volume at top dead center. When the parameters in the heat loss
equation are normalized by the conditions at state 1, bottom dead center, they take the form
𝑄̃ =
𝑄in
𝑃1 𝑉1
𝒉̃ =
4𝒉𝑇1
𝑃1 𝜔𝛽𝑏
𝑄l
𝑃1 𝑉1
(2.48)
4𝑉1
𝑏(𝐴o − 4𝑉o ∕𝑏)
(2.49)
𝑇
𝑇̃ =
𝑇1
𝑄̃ l =
and
𝛽=
The dimensionless heat loss is then
̃l
𝒉𝐴𝑇1
𝑑𝑄
̃ + 𝛽 𝑉̃ )(𝑃̃ 𝑉̃ − 𝑇̃w )
=
(𝑇̃ − 𝑇̃w ) = 𝒉(1
𝑑𝜃
𝑃1 𝑉1 𝜔
(2.50)
We can express the volume term 𝛽 as a function of the compression ratio 𝑟. Since
𝑟 = 𝑉1 ∕𝑉o ,
𝛽=
4𝑟
𝑏(𝐴o ∕𝑉o ) − 4
(2.51)
For example, for a square engine (bore 𝑏 = stroke 𝑠) with a flat top piston and cylinder head
geometry,
𝐴o ∕𝑉o =
2(𝑟 − 1) + 4
𝑏
(2.52)
Finite Energy Release
51
and
2𝑟
(2.53)
𝑟−1
Note that when heat transfer losses are added, there are additional dependencies on the
dimensionless wall temperature, heat transfer coefficient, and compression ratio.
If the mass in the cylinder is no longer constant due to blowby, the logarithmic
derivative of the equation of state becomes
𝛽=
1 𝑑𝑉
1 𝑑𝑚 1 𝑑𝑇
1 𝑑𝑃
+
=
+
(2.54)
𝑃 𝑑𝜃
𝑉 𝑑𝜃
𝑚 𝑑𝜃
𝑇 𝑑𝜃
Similarly, the first law of thermodynamics in differential form applicable to an open
system must be used.
𝑑𝑄
𝑑𝑚 𝑚̇ l ℎl
𝑑𝑇
𝑑𝑉
−𝑃
= 𝑚𝑐𝑣
+ 𝑐𝑣 𝑇
+
(2.55)
𝑑𝜃
𝑑𝜃
𝑑𝜃
𝑑𝜃
𝜔
The term 𝑚̇ l is the instantaneous rate of leakage or blowby flow. The enthalpy of the blowby
is assumed to the same as that of the cylinder, so ℎl = 𝑐𝑝 𝑇 .
From the mass conservation equation applied to the cylinder,
𝑚̇
𝑑𝑚
=− l
𝑑𝜃
𝜔
Eliminating 𝑑𝑇 ∕𝑑𝜃 between Equations 2.54 and 2.55 yields the following:
(2.56)
(𝛾 − 1) 𝑑𝑄 𝛾 𝑚̇ l
𝑃 𝑑𝑉
𝑑𝑃
= −𝛾
+
−
𝑃
(2.57)
𝑑𝜃
𝑉 𝑑𝜃
𝑉
𝑑𝜃
𝜔𝑚
Including heat transfer loss as per Equation 2.45 and defining the blowby coefficient
𝐶 as
𝑚̇
𝐶= l
(2.58)
𝑚
results in the following four ordinary differential equations for pressure, work, heat loss,
and cylinder mass as a function of crank angle:
] 𝛾𝐶 𝑃̃
(𝛾 − 1) [ ̃ 𝑑𝑥 ̃
𝑃̃ 𝑑 𝑉̃
𝑑 𝑃̃
= −𝛾
+
− 𝒉(1 + 𝛽 𝑉̃ )(𝑃̃ 𝑉̃ ∕𝑚̃ − 𝑇̃w ) −
𝑄
𝑑𝜃
𝑑𝜃
𝜔
𝑉̃ 𝑑𝜃
𝑉̃
̃
𝑑 𝑉̃
𝑑𝑊
= 𝑃̃
𝑑𝜃
𝑑𝜃
(2.59)
𝑑 𝑄̃ l
̃ + 𝛽 𝑉̃ )(𝑃̃ 𝑉̃ ∕𝑚̃ − 𝑇̃w )
= 𝒉(1
𝑑𝜃
𝑚̃
𝑑 𝑚̃
= −𝐶
𝑑𝜃
𝜔
The above four linear equations are solved numerically in the MATLABⓇ program
FiniteHeatMassLoss.m, which is listed in Appendix F. The program is a finite
energy release program that can be used to compute the performance of an engine and
includes both heat and mass transfer. The engine performance is computed by numerically
integrating Equation 2.59 for the pressure, work, heat loss, and cylinder gas mass as
a function of crank angle. The integration starts at bottom dead center (𝜃 = −180◦ ),
with initial inlet conditions given. The integration proceeds degree by degree to top
dead center and back to bottom dead center. Once the pressure and other terms are
computed as a function of crank angle, the overall cycle parameters of net work, thermal
52
Heat Engine Cycles
efficiency, and imep are also computed. The use of the program is detailed in the following
example.
EXAMPLE 2.3
Finite Energy Release with Heat and Mass Loss
For the same engine conditions as in Example 2.2, find the maximum imep and thermal
efficiency when heat and mass loss is accounted for. Vary the start of energy release from
𝜃s = −50◦ to 𝜃s = +20◦ atdc. Assume the heat transfer coefficient 𝒉 = 500 W(m2 K), the
cylinder wall temperature 𝑇w = 360 K, the top dead center area/volume ratio 𝐴o ∕𝑉o =
306.6 m−1 , and the mass transfer parameter 𝐶 = 0.08 s−1 , with engine speed 𝜔 = 200 rad/s.
̃ and 𝑇̃w for this problem are
SOLUTION The non dimensional parameters 𝛽, 𝒉,
𝛽=
𝒉̃ =
(4)(10)
4𝑟
=
= 1.50
𝑏(𝐴o ∕𝑉o ) − 4 (0.1)(306.6) − 4
4𝒉𝑇1
(4)(500)(300)
=
= 0.20
𝑃1 𝜔𝛽𝑏 (100, 000)(200)(1.5)(0.1)
𝑇̃w = 𝑇w ∕𝑇1 = 360∕300 = 1.2
The above engine parameters are entered into the FiniteHeatMassLoss.m program as shown below.
function [ ] = FiniteHeatMassLoss( )
Gas cycle heat release code with heat and mass transfer
thetas = - 20;
start of heat release
(deg)
thetad = 40;
duration of heat release
r = 10;
compression ratio
(deg)
gamma = 1.4;
ideal gas const
Q = 20.;
dimensionless total heat release
h = 0.2;
dimensionless heat transfer coeff.
tw = 1.2;
dimensionless cylinder wall temp
beta = 1.5;
dimensionless volume
a = 5;
Wiebe parameter
n = 3;
Wiebe exponent
omega = 200.;
engine speed
c = 0.8;
mass loss coefficient
...
The results are presented in Figures 2.15 and 2.16, and representative thermodynamic
parameters are compared with the simple energy release computation with no heat or mass
loss in Table 2.1. With the heat and mass transfer included, maximum efficiency is reduced
from 0.60 to 0.52, and the maximum nondimensional imep is reduced from 13.24 to 11.55.
The general dependence of the efficiency and imep on the start of energy release is very
similar for both cases, as the optimum start of ignition remains at −20◦ and the peak
pressure crank angle remains at +11◦ .
Finite Energy Release
0.65
53
w/o heat and mass loss
w/ heat and mass loss
Thermal efficiency η
0.6
0.55
0.5
0.45
0.4
0.35
Figure 2.15 Thermal efficiency
versus start of energy release for
Examples 2.2 and 2.3.
0.3
–50
–40
–30
–20
–10
0
Start of heat release, θs (deg)
10
20
10
20
16
w/o heat and mass loss
w/ heat and mass loss
Imep/P1
14
12
10
8
Figure 2.16 Imep versus start of
energy release for Examples 2.2
and 2.3.
6
–50
–40
–30
–20
–10
0
Start of heat release, θs (deg)
Table 2.1 Comparison of Energy Release Models With and Without
Heat/Mass Transfer Loss at 𝜃s = −20◦ and 𝜃d = 40◦
𝑃max ∕𝑃1
𝜃max
Net work/𝑃1 𝑉1
Efficiency 𝜂
𝜂∕𝜂Otto
Imep/𝑃1
Without heat and
mass loss
With heat and
mass loss
87.77
11.00
11.91
0.596
0.990
13.24
85.31
11.00
10.39
0.520
0.863
11.55
54
Heat Engine Cycles
Cumulative work and heat loss
12
10
Work
Heat loss
8
6
4
2
0
–2
Figure 2.17 Cumulative work
and heat/mass loss for
Example 2.3.
–4
–150
–100
–50
0
50
100
150
Crank angle θ (deg)
The cumulative work and heat/mass transfer loss is plotted in Figure 2.17 as a function
of crank angle for the optimum case of 𝜃s = −20◦ . The cumulative work is initially negative
due to the piston compression and becomes positive on the expansion stroke. The heat
transfer loss is very small during compression, indicating a nearly isentropic compression
process, and is somewhat linear during the expansion process.
2.7 IDEAL FOUR-STROKE PROCESS AND RESIDUAL FRACTION
The simple gas cycle models assume that the heat rejection process occurs at constant volume, and neglect the gas flow that occurs when the intake and exhaust valves
are opened and closed. In this section, we use the energy equation to model the exhaust and intake strokes, and determine the residual fraction of gas remaining in the
cylinder.
At this level of modeling, we need to make some assumptions about the operation
of the intake and exhaust valves. During the exhaust stroke, the exhaust valve is assumed
to open instantaneously at bottom dead center and close instantaneously at top dead center. Similarly, during the intake stroke, the intake valve is assumed to open at top dead
center and remain open until bottom dead center. The intake and exhaust valve overlap, that is, the time during which they are open simultaneously, is therefore assumed to
be zero.
The intake and exhaust strokes are also assumed to occur adiabatically and at constant
pressure. Constant pressure intake and exhaust processes occur only at low engine speeds.
More realistic computations model the instantaneous pressure drop across the valves and
furthermore would account for the heat transfer that is especially significant during the
exhaust. Such considerations are deferred to Chapters 5 and 9.
Referring to Figure 2.18, the ideal intake and exhaust processes are as follows:
4 to 5a
5a to 6
6 to 7
7 to 1
Constant cylinder volume blowdown
Constant pressure exhaustion
Constant cylinder volume reversion
Constant pressure induction
Ideal Four-Stroke Process and Residual Fraction
4
4
Pi, Pe
6, 7
55
1, 5a
5
Pe
Pi
Unthrottled cycle
6
5a
5
1
7
Throttled cycle
4
Pi
Pe
7
1
5a
6
5
Supercharged cycle
Figure 2.18 Four-stroke inlet and exhaust flow. 𝑃i = inlet pressure; 𝑃e = exhaust pressure.
Exhaust Stroke
The exhaust stroke has two processes: gas blowdown and gas displacement. At the end of
the expansion stroke 3 to 4, the pressure in the cylinder is greater than the exhaust pressure.
Hence, when the exhaust valve opens, gas will flow out of the cylinder even if the piston
does not move. Typically, the pressure ratio, 𝑃4 ∕𝑃e , is large enough to produce sonic flow at
the valve so that the pressure in the cylinder rapidly drops to the exhaust manifold pressure,
𝑃e , and the constant volume approximation is justified. The remaining gas in the cylinder
that has not flowed out through the exhaust valve undergoes an expansion process. If heat
transfer is neglected, this unsteady expansion process can be modeled as isentropic. Note
that both the closed valve expansion from 3 to 4 and the open valve expansion from 4 to 5
are modeled as isentropic processes.
Therefore, the temperature and pressure of the exhaust gases remaining in the cylinder
are
( )(𝛾−1)∕𝛾
𝑃5
(2.60)
𝑇5 = 𝑇4
𝑃4
𝑃5 = 𝑃e
(2.61)
As the piston moves upward from bottom dead center, it pushes the remaining cylinder
gases out of the cylinder. The cylinder pressure is assumed to remain constant at 𝑃5 = 𝑃6 =
𝑃4 . Since internal combustion engines have a clearance volume, not all of the gases will
be pushed out. There will be exhaust gas left in the clearance volume, called residual gas.
This gas will mix with the incoming air or fuel--air mixture, depending on the location of
the fuel injectors.
The state of the gas remaining in the cylinder during the exhaust stroke can be found
by applying the closed system first law to the cylinder gas from state 5 to state 6 as shown
in Figure 2.19. The closed system control volume will change in shape as the cylinder
gases flow out of the exhaust port across the exhaust valve. Note that while the blowdown
is assumed to occur at constant cylinder volume, the control mass is assumed to expand
isentropically.
The energy equation is
𝑄5−6 − 𝑊5−6 = 𝑈6 − 𝑈5
(2.62)
Heat Engine Cycles
Isentropic expansion
Pressure
56
4
Pe
6
5
5a
Volume
Intake
valve
Exhaust
valve
State 4
Bottom dead center
Intake
valve
Exhaust
valve
Intake
valve
State 5
Bottom dead center
Exhaust
valve
State 6
Top dead center
Control mass is shaded
Figure 2.19 The exhaust stroke (4 to 5 to 6) illustrating residual mass.
The work term is
𝑊5−6 = 𝑃e (𝑉6 − 𝑉5 )
(2.63)
and if the flow is assumed to be adiabatic, the first law becomes
𝑈6 + 𝑃e 𝑉6 = 𝑈5 + 𝑃e 𝑉5
(2.64)
ℎ6 = ℎ5
(2.65)
𝑇e = 𝑇 6 = 𝑇 5
(2.66)
or
Therefore, during an adiabatic exhaust stroke, the enthalpy and temperature of the
exhaust gases remain constant as they leave the cylinder, and the enthalpy of the residual
gas left in the cylinder clearance volume is constant. The residual gas fraction, 𝑓 , is the
Ideal Four-Stroke Process and Residual Fraction
57
ratio of the residual gas mass, 𝑚r = 𝑚6 , in the cylinder at the end of the exhaust stroke
(state 6) to the mass, 𝑚 = 𝑚1 = 𝑚4 , of the fuel--air mixture:
𝑓=
𝑉6 ∕𝑣6
1 𝑣4
1 𝑇4 𝑃e
=
=
𝑉4 ∕𝑣4
𝑟 𝑣6
𝑟 𝑇e 𝑃 4
(2.67)
Since
𝑇e = 𝑇4
(
𝑃e
𝑃4
)(𝛾−1)∕𝛾
(2.68)
)1∕𝛾
(2.69)
the residual fraction is
1
𝑓=
𝑟
(
𝑃e
𝑃4
For example, for a compression ratio of 𝑟 = 9, 𝑃e = 101 kPa, 𝑃4 = 500 kPa, and
𝛾 =1.3, 𝑓 = 1∕9(101∕450)1∕1.3 = 0.035. Typical values of the residual gas fraction, 𝑓 , are
in the 0.03--0.12 range. The residual gas fraction is lower in Diesel cycle engines than in
Otto cycle engines, due to the higher compression ratio in Diesel cycles.
Intake Stroke
When the intake valve is opened, the intake gas mixes with the residual gas. Since the
intake gas temperature is usually less than the residual gas temperature, the cylinder gas
temperature at the end of the intake stroke will be greater than the intake temperature. In
addition, if heat transfer is neglected, the flow across the intake valve, either from the intake
manifold to the cylinder or the reverse, is at constant enthalpy.
There are three different flow situations for the intake stroke, depending on the ratio of
inlet to exhaust pressure. If the inlet pressure is less than the exhaust pressure, the engine
is throttled. In this case, there is flow from the cylinder into the intake port when the intake
valve opens. In the initial portion of the intake stroke, the induced gas is primarily composed
of combustion products that have previously flowed into the intake port. In the latter portion
of the stroke, the mixture flowing in is fresh charge, undiluted by any combustion products.
If the inlet pressure is greater than the exhaust pressure, the engine is said to be
supercharged (turbocharging is a special case of supercharging in which a compressor
driven by an exhaust turbine raises the pressure of atmospheric air delivered to an engine).
In this case, there is flow from the intake port into the engine until the pressure equilibrates.
In actual engines, because of valve overlap, there may be a flow of fresh mixture from
the inlet to the exhaust port, which can waste fuel and be a source of hydrocarbon exhaust
emissions. The third case is when inlet and exhaust pressures are equal; the engine is then
said to be unthrottled.
The unsteady open system mass and energy equations can be used to determine the
state of the fuel--air mixture and residual gas combination at state 1, the end of the intake
stroke. The initial state of the gas in the system at the beginning of the intake process is at
state 6.
As discussed above, there is a flow of a gas mixture into or out of the cylinder when the
intake valve is opened, depending on the relative pressure difference. The net gas flow into
the cylinder control volume has mass 𝑚i , enthalpy ℎi , and pressure 𝑃i . As the piston moves
downward, it is assumed that the cylinder pressure remains constant at the inlet pressure
𝑃i , which is consistent with experimental observations. For the overall process from state
58
Heat Engine Cycles
6 to state 1 with the inlet flow at state ‘‘i’’, the conservation of mass equation is
𝑚i = 𝑚1 − 𝑚6
(2.70)
The unsteady energy equation is
𝑄6−1 − 𝑊6−1 = −𝑚i ℎi + 𝑚1 𝑢1 − 𝑚6 𝑢6
(2.71)
If heat transfer is neglected, 𝑄6−1 = 0, and the work done by the gas is 𝑊6−1 =
𝑃1 (𝑉1 − 𝑉6 ), so
−𝑃i (𝑉1 − 𝑉6 ) = −(𝑚1 − 𝑚6 )ℎi + 𝑚1 𝑢1 − 𝑚6 𝑢6
(2.72)
Since 𝑢1 = ℎ1 − 𝑃1 𝑣1 and 𝑢6 = ℎ6 − 𝑃e 𝑣6 , we can write the energy equation in terms
of enthalpy:
(𝑃i − 𝑃e )𝑚6 𝑣6 = −(𝑚1 − 𝑚6 )ℎi + 𝑚1 ℎ1 − 𝑚6 ℎ6
Solving for ℎ1 ,
ℎ1 =
)
(
[
]
𝑚6
𝑚1
− 1 ℎi + (𝑃i − 𝑃e )𝑣6
ℎ6 +
𝑚1
𝑚6
(2.73)
(2.74)
Therefore, the enthalpy at the end of the intake stroke is not just the average of the
initial and intake enthalpies, as would be the case for a steady flow situation, but also
includes the flow work term.
The equation for the enthalpy at the end of the intake stroke, Equation 2.74, can also
be expressed in terms of the residual gas fraction, 𝑓 . From Equation 2.67,
𝑚6 = 𝑚1 𝑓
and
𝑚1 − 𝑚6 = 𝑚1 (1 − 𝑓 )
(2.75)
so
𝑚i = 𝑚(1 − 𝑓 )
(2.76)
𝑃e 𝑣6 = 𝑅 𝑇6
(2.77)
and from the ideal gas law,
Upon substitution of Equations 2.76 and 2.77 into Equation 2.74,
(
)
𝑃i
ℎ1 = (1 − 𝑓 )ℎi + 𝑓 ℎe − 1 −
𝑓 𝑅 𝑇e
𝑃e
If the reference enthalpy is chosen so that ℎi = 𝑐𝑝 𝑇i , then
[
(
)(
)]
𝑃
𝛾 −1
𝑇1 = (1 − 𝑓 )𝑇i + 𝑓 1 −
1− i
𝑇e
𝛾
𝑃e
(2.78)
(2.79)
For example, if 𝑓 = 0.05, 𝑃i ∕𝑃e = 0.5, 𝛾 = 1.35, 𝑇i = 320 K, and 𝑇e = 1400 K, then
𝑇1 = 365 K.
The volumetric efficiency of the inlet stroke for a gas cycle is given by
𝑒𝑣 =
𝑃 ∕𝑃 − 1
𝑚i
=1− e i
𝜌i 𝑉 d
𝛾(𝑟 − 1)
(2.80)
During the intake process, the gas within the control volume does work since the piston
is expanding the cylinder volume. During exhaust, work is done on the gas. The net effect
during the intake and exhaust strokes is
𝑊5a−1 = (𝑃i − 𝑃e )𝑉d
(2.81)
Ideal Four-Stroke Process and Residual Fraction
59
The negative of that work is called pumping work since it is a loss of useful work for
the throttled engine. The pumping mean effective pressure is defined as the pumping work
per unit displacement volume:
(2.82)
pmep = 𝑃e − 𝑃i
The indicated mean effective pressure (imep) is defined as the work per unit displacement volume done by the gas during the compression and expansion strokes. The work
per unit displacement volume required to pump the working fluid into and out of the engine during the intake and exhaust strokes is termed the pumping mean effective pressure
(pmep). It is the sum of the pressure drops across flow restrictions during the intake and
exhaust strokes, including intake system, valves, and the exhaust system.
The following relations should be clear:
(imep)net = imep − pmep
)
(
imep
𝜂net = 𝜂 1 −
pmep
(2.83)
(2.84)
Four-Stroke Otto Gas Cycle Analysis
When we include the exhaust and intake strokes, we have two additional equations for
the gas cycle analysis, the exhaust energy equation, and the intake energy equation. The
two unknown parameters in these equations are the residual gas fraction, 𝑓 , and the gas
temperature at the end of the intake stroke, 𝑇1 . When the residual gas fraction 𝑓 is taken
into account, the heat addition, 𝑄in , is
(2.85)
𝑄in = 𝑚i 𝑞in = 𝑚(1 − 𝑓 )𝑞in
where 𝑞in is the heat addition per unit mass of gas inducted.
The cycle input parameters in this four-stroke gas cycle analysis are summarized in
Table 2.2. Since it is difficult to solve these two equations algebraically, the solution is
found by iteration, as shown in this section. Since 𝑇1 is dependent on the residual gas fraction 𝑓 and the residual gas temperature 𝑇e , we first need to estimate the values of 𝑓 and 𝑇e ,
and then iterate through the cycle calculation repeatedly to get converged values of 𝑓 and 𝑇e .
6, i -1: Intake stroke
[
𝑇1 = (1 − 𝑓 )𝑇i + 𝑓 1 −
(
𝛾 −1
𝛾
)(
𝑃
1− i
𝑃e
Table 2.2 Input Parameters for Four-Stroke Gas Cycle
Parameter
Description
𝑇i
𝑟
𝑃e
𝑃i
𝛾
𝑞n
Inlet air or mixture temperature
Compression ratio
Exhaust pressure
Inlet pressure
Ideal gas specific heat ratio
Heat added per unit mass of gas induced
)]
𝑇e
60
Heat Engine Cycles
𝑃1 = 𝑃i
1-2: Isentropic compression stroke
𝑃2 = 𝑃1 (𝑉1 ∕𝑉2 )𝛾 = 𝑃1 𝑟𝛾
𝑇2 = 𝑇1 𝑟𝛾−1
2-3: Constant volume heat addition
𝑇3 = 𝑇2 + 𝑞in (1 − 𝑓 )∕𝑐𝑣
𝑃3 = 𝑃2 (𝑇3 ∕𝑇2 )
3-4: Isentropic expansion stroke
𝑃4 = 𝑃3 (1∕𝑟)𝛾
𝑇4 = 𝑇3 (1∕𝑟)𝛾−1
4-5: Isentropic blowdown
𝑇5 = 𝑇4 (𝑃4 ∕𝑃e )(1−𝛾)∕𝛾
𝑃5 = 𝑃e
5-6: Constant pressure adiabatic exhaust stroke
𝑇e = 𝑇5
𝑃6 = 𝑃5 = 𝑃e
𝑓 = 1∕𝑟(𝑃6 ∕𝑃4 )1∕𝛾
Appendix F contains a listing of the program FourStrokeOtto.m that iterates
through the above four-stroke Otto gas cycle equations to determine the cycle pressures,
temperatures, and the overall thermal parameters.
EXAMPLE 2.4
Four-Stroke Otto Cycle
Compute the volumetric efficiency, net thermal efficiency, residual fraction, intake stroke
temperature rise, 𝑇1 − 𝑇i , and the exhaust stroke temperature decrease, 𝑇4 − 𝑇e , of an
engine that operates on the ideal four-stroke Otto cycle. The engine is throttled with an
inlet pressure of 𝑃i = 50 kPa and has an inlet temperature of 𝑇i = 300 K. The exhaust
pressure is 𝑃e = 100 kPa. The compression ratio 𝑟 = 10. Assume a heat input of 𝑞in =
2500 kJ/kg and 𝛾 = 1.3. Plot the volumetric efficiency, net thermal efficiency, and residual
fraction as a function of the intake/exhaust pressure ratio for 0.3 < 𝑃i ∕𝑃e < 1.5.
SOLUTION
Ideal Four-Stroke Process and Residual Fraction
61
The program input portion of FourStrokeOtto.m is shown below.
Four-stroke Otto cycle model
Input parameters:
Ti = 300;
inlet temperature (K)
Pi = 50;
inlet pressure (kPa)
Pe = 100;
exhaust pressure (kPa)
r = 10;
compression ratio
qin = 2500;
heat input, kJ/kg(gas)
R = 0.287;
gas constant (kJ/kg K)
f = 0.05;
guess value of residual fraction f
Tr = 1000;
guess value of exhaust temp (K)
tol = 0.001; convergence tolerance
....
For the above conditions, as shown in Tables 2.3 and 2.4, the computation indicates
that the intake stroke temperature rise, 𝑇1 − 𝑇i , is about 45 K and the exhaust blowdown
temperature decrease, 𝑇4 − 𝑇e , is about 280 K. The volumetric efficiency 𝑒𝑣 = 0.91, the
net thermal efficiency 𝜂 = 0.46, and the residual fraction 𝑓 = 0.053.
The volumetric efficiency, Equation 2.80, the residual fraction, Equation 2.69, and
the net thermal efficiency, Equation 2.84, are plotted in Figures 2.20, 2.21, and 2.22,
respectively, as a function of the intake/exhaust pressure ratio.
Comment: As the pressure ratio increases, the volumetric efficiency and thermal efficiency
increase, and the residual fraction decreases. The dependence of the volumetric efficiency
𝑒𝑣 on compression ratio is reversed for the throttled and supercharged conditions. In
addition, the residual gas fraction increases. The increase in residual fraction is due to
the decrease in the intake mass relative to the residual mass as the intake pressure is
decreased.
Table 2.3 State Variables for Four-Stroke Gas Cycle Example 2.4
State
Pressure (kPa)
Temperature (K)
1
50.0
345.3
2
997.6
688.9
3
4582.6
3164.3
4
229.7
1585.9
Table 2.4 Cycle Parameters for Four-Stroke Gas Cycle Example 2.4
Residual fraction 𝑓
Net imep (kPa )
Ideal thermal efficiency 𝜂
Net thermal efficiency 𝜂net
Exhaust temperature (K)
Volumetric efficiency 𝑒𝑣
0.053
612.0
0.499
0.461
1309
0.91
62
Heat Engine Cycles
Volumetric efficiency ( ev )
1.05
1
r = 15
0.95
10
0.9
0.85
0.8
0.3
Figure 2.20 Volumetric
efficiency for Example 2.3.
0.5
Throttled
Supercharged
0.7
1.1
0.9
1.3
1.5
Intake/exhaust pressure ratio
0.08
Residual fraction ( f )
0.07
0.06
0.05
0.04
r = 10
0.03
0.02
15
0.01
Figure 2.21 Residual fraction
for Example 2.3.
0
0.3
0.5
0.7
0.9
1.1
1.3
1.5
Intake/exhaust pressure ratio
2.8 DISCUSSION OF GAS CYCLE MODELS
Maximizing the mean effective pressure is important in engine design so that one can
build a smaller, lighter engine to produce a given amount of work. As shown in Equation
2.11, there are evidently two ways to do this: (1) by increasing the compression ratio 𝑟
and (2) by increasing the heat input 𝑄in . However, there are practical limitations to these
approaches. For spark ignition engines of conventional design, the compression ratio must
be low enough to avoid engine knock, whereas for diesel engines increasing engine friction
limits the utility of increasing compression ratio. Other more complicated factors influence
the selection of compression ratio, especially constraints imposed by emission standards
and, for some diesel engines, problems of startability.
One might expect that we can increase 𝑄in by increasing the fuel flow rate delivered to
an engine. As we shall see in our studies of fuel--air cycles in Chapter 4, this is not always
References
63
0.6
Net thermal efficiency
r = 15
0.55
10
0.5
0.45
Figure 2.22 Net thermal
efficiency for Example 2.3.
0.4
0.3
0.5
0.7
0.9
1.1
1.3
Intake/exhaust pressure ratio
1.5
correct. With fuel-rich mixtures not all of the fuel energy is used, since there is not enough
oxygen to burn the carbon monoxide to carbon dioxide nor the hydrogen to water. The
fuel--air cycle predicts that the efficiency decreases as the mixture is made richer beyond
stoichiometric.
According to the gas cycles, and the fuel--air cycles to be discussed later, the efficiency
is greatest if heat can be added at constant volume:
𝜂Otto > 𝜂dual > 𝜂Diesel
(2.86)
Why then do we build engines that resemble constant pressure heat addition when we
recognize that constant volume heat addition would be better? To illustrate how difficult
that question is let us ask the following: Suppose that the maximum pressure in the cycle
must be less than some value 𝑃max . How should the heat be added to produce the required
work? The answer is now
𝜂Diesel > 𝜂dual > 𝜂Otto
(2.87)
This can be demonstrated with the aid of a temperature--entropy diagram. If the Otto
cycle and the Diesel cycle are drawn on such a diagram so that the work done in each cycle
is the same, it can then be shown (as per the homework problem at the end of the chapter)
that the Diesel cycle is rejecting less heat and must therefore be the most efficient.
2.9 REFERENCES
GHOJEL, J. (2010), ‘‘Review of the Development and Applications of the Wiebe Function,’’
Int. J. Eng. Res., Vol. 11, pp. 297--312.
FOSTER, D. (1985), ‘‘An Overview of Zero-Dimensional Thermodynamic Models for IC
Engine Data Analysis,’’ SAE Technical Paper 852070.
HEYWOOD, J. B. (1988), Internal Combustion Engine Fundamentals, Mc-Graw-Hill,
New York.
MILLER, R. H. (1947), ‘‘Supercharging and Internal Cooling Cycle for High Output,’’ ASME
Trans., Vol. 69, pp. 453--464.
64
Heat Engine Cycles
2.10 HOMEWORK
2.1
An engine cylinder contains 7 × 10−5 kg of fuel with a heat of combustion, 𝑞c , of 45,000
kJ/kg. The volume 𝑉1 at top dead center is 0.15 × 10−3 m3 , and the volume 𝑉2 at bottom
dead center is 1.50 × 10−3 m3 . The air--fuel ratio is 16:1, and the mixture temperature 𝑇1 at
the start of compression is 300 K. Modeling the compression and combustion as an ideal
gas (𝛾 = 1.4, 𝑐𝑣 = 0.87 kJ/(kg K)) Otto cycle, (a) what is the maximum temperature 𝑇3 and
pressure 𝑃3 ? and (b) what is the pressure 𝑃1 at the start of compression?
2.2
The Lenoir air cycle is composed of three processes: 1-2 constant volume heat addition,
2-3 isentropic expansion, and 3-1 constant pressure heat rejection. This cycle is named
after Jean Lenoir (1822--1900), a Belgian engineer who developed an internal combustion
engine in 1858. It is a cycle in which combustion occurs without compression of the
mixture. (a) Draw the Lenoir cycle on 𝑝--𝑉 and 𝑇 --𝑠 diagrams. (b) Assuming the working
fluid is an ideal gas with constant properties, derive an expression for the thermal efficiency
of the Lenoir air cycle. (c) Compare the Lenoir cycle thermal efficiency to the Otto cycle
efficiency for the standard inlet conditions, 𝑄in = 1000 J, 𝑟 = 8, 𝑚 = 1.0 g, and 𝛾 = 1.4.
2.3 (a) Show for an Otto cycle that 𝑇3 ∕𝑇2 = 𝑇4 ∕𝑇1 .
(b) Derive the Otto cycle efficiency equation, Equation 2.10.
2.4
Derive the Otto and Diesel cycle imep equation, Equation 2.11.
2.5
For equal maximum temperature and heat input, which cycle will be more efficient, the
Diesel or Otto? Prove your answer by comparing the two cycles on the 𝑇 --𝑠 diagram. The
two cycles should have a common state corresponding to the start of compression.
2.6 (a) Show that for a Diesel cycle (𝑇3 ∕𝑇2 )𝛾 = 𝑇4 ∕𝑇1 .
(b) Derive Equations 2.15 and 2.16.
2.7
What does the compression ratio of a Diesel cycle need to be to have the same thermal
efficiency of an Otto cycle engine that has a compression ratio 𝑟 = 9? Assume the specific
heat ratio 𝛾 = 1.3 and 𝑄in ∕𝑃1 𝑉1 = 30.
2.8
A Diesel cycle has a compression ratio of 20, and the heat input 𝑞in to the working fluid
is 1600 kJ/kg. The Diesel cycle is unthrottled, so at the start of compression 𝑃1 = 101 kPa
and 𝑇1 = 298 K. Assuming the working fluid is air with constant properties, what is the
maximum pressure and temperature in the cycle, the cycle efficiency, and imep?
2.9
Show that for the Otto cycle as 𝑟 → 1, imep/𝑃1 → 𝑄in (𝛾 − 1)∕𝑃1 𝑉1 (use l’Hopital’s rule).
2.10
A engine is modeled with a limited pressure cycle. The maximum pressure is to be 8000 kPa.
The compression ratio is 17:1, the inlet conditions are 101 kPa and 320 K, and the nondimensional heat input 𝑄in ∕𝑃1 𝑉1 = 30. Find the thermal efficiency and the values of 𝛼
and 𝛽. Assume 𝛾 = 1.3.
2.11 (a) Derive the equation for the Miller cycle efficiency, Equation 2.23.
(b) Derive the equation for the Miller cycle imep, Equation 2.24.
2.12
For Otto and Miller cycles that have equal compression ratios (𝑟𝑐 = 10), what are the
respective thermal efficiencies and non dimensional imeps? Assume that the parameter 𝜆
is equal to 1.5 for the Miller cycle, the specific heat ratio 𝛾 = 1.3, and 𝑄in ∕𝑃1 𝑉1 = 30.
Homework
65
2.13
Develop a complete expansion cycle model in which the expansion stroke continues until the
pressure is atmospheric. Derive an expression for the efficiency in terms of 𝛾, 𝛼 = 𝑉4 ∕𝑉3 ,
and 𝛽 = 𝑉1 ∕𝑉4 .
2.14
If a four-cylinder, four-stroke engine with a 0.1 m bore and a 0.08 m stroke operating at
2000 rpm has the same heat/mass loss parameters as Example 2.3, how much indicated
power (kW) would it produce? What if it were a two-stroke engine?
2.15
Using the program BurnFraction.m, and assuming that 𝑎 = 5, the beginning of heat
addition is −10◦ , and the duration of heat addition is 40◦ , (a) plot the Wiebe heat release
fraction curve for the following form factor values: 𝑛 = 2, 3, and 4. (b) At what crank angle
is 0.10, 0.50, and 0.90 of the heat released?
2.16
Using the program FiniteHeatRelease.m, determine the effect of heat release duration on the net work, power, mean effective pressure, and thermal efficiency for a four-stroke
engine with heat release durations of 40◦ , 30◦ , 20◦ , 10◦ , and 5◦ . Assume that the total heat
addition 𝑄in = 2500 J, the start of heat release 𝜃s remains constant at −10◦ atdc, 𝑎 = 5,
𝑛 = 3, and 𝛾 = 1.4. The engine bore and stroke are 0.095 m, the compression ratio is 9:1,
and engine speed is 3000 rpm.
2.17
If a four-cylinder unthrottled Otto cycle engine is to generate 100 kW at an engine speed
of 2500 rpm, what should its bore and stroke be? Assume a square block engine with equal
bore and stroke and a compression ratio of 10:1. The total heat addition 𝑄in = 2200 J,
the start of heat release 𝜃s remains constant at −15◦ atdc, the combustion duration is 40◦ ,
𝑎 = 5, 𝑛 = 3, and 𝛾 = 1.4. Use the single-cylinder program FiniteHeatRelease.m
and solve for a power output of 25 kW.
2.18
Develop a four-stroke Diesel cycle model (along the lines used in Example 2.4) with
the following data: 𝑟 = 22, 𝛾 = 1.3, 𝑇i = 300 K, 𝑃i = 101 kPa, 𝑃i ∕𝑃e = 0.98, 𝑀 = 29, and
𝑞in = 2090 kJ/kggas .
2.19
Using the program FourStrokeOtto.m, plot the effect of inlet throttling from 100 to
25 kPa on the peak pressure, 𝑃3 , and the volumetric efficiency 𝜂v . Assume the following
conditions: 𝑇i = 300 K, 𝑟 = 9, 𝛾 = 1.3, and 𝑞in = 2400 kJ/kggas .
2.20
In Example 2.3, 𝑇e is the exhaust temperature during the constant pressure exhaust stroke.
It is not the same as the average temperature of the gases exhausted. Explain.
Chapter
3
Fuel, Air, and Combustion
Thermodynamics
3.1 INTRODUCTION
It has already been mentioned that an understanding of internal combustion engines will
require a better thermodynamic model than the ideal gas models used in Chapter 2. In
this chapter, we review the thermodynamics of combustion and develop models suitable
for application to internal combustion engines. The chapter begins with multicomponent
ideal gas property models, followed by stoichiometry, and then computation of equilibrium
combustion components and properties. We will develop equations for the thermodynamic
properties of fuel--air--residual gas mixtures as a function of the pressure, temperature, and
the mole fractions of the component species, and introduce the equivalence ratio.
A few words about the atmosphere are in order. The properties of air vary geographically, with altitude, and with time. In this text, we will assume that air is 21% oxygen and
79% nitrogen by volume, that is for each mole of O2 , there are 3.76 moles of N2 . Selected
physical properties of air, oxygen, and nitrogen are given in Appendices A and B. Extension
of our analyses to different air mixtures encountered in practice is straightforward. The
most frequent differences accounted for are the presence of water and argon in air.
3.2 THERMODYNAMIC PROPERTIES OF IDEAL GAS MIXTURES
In computing cycle parameters, thermal efficiency, and work produced by an engine, we
need to compute the changes of state due to combustion, isentropic compression/expansion,
and blowdown. In the analyses that follow, we model the fuel--air mixture and the products
of combustion as ideal gas mixtures. For an ideal gas, the familiar relationships between
pressure 𝑃 , temperature 𝑇 , and volume 𝑉 are
𝑃 𝑉 = 𝑁 𝑅u 𝑇
𝑃𝑉 = 𝑚𝑅𝑇
(3.1)
𝑃𝑣 = 𝑅𝑇
The mass, 𝑚 (kg), of a gas mixture is the sum of the mass of all 𝑛 components
𝑚=
𝑛
∑
𝑚𝑖
(3.2)
𝑖=1
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
66
Thermodynamic Properties of Ideal Gas Mixtures
67
The mass fraction, 𝑥𝑖 , of any given species is defined as
𝑥𝑖 = 𝑚𝑖 ∕𝑚
(3.3)
and it should be clear that
𝑛
∑
𝑥𝑖 = 1
(3.4)
𝑖=1
The total number of moles, 𝑁, of a mixture is the sum of moles of all 𝑛 components
𝑁=
𝑛
∑
𝑛𝑖
(3.5)
𝑖=1
and the mole fraction 𝑦𝑖 of any given species is the fraction of the total number of moles
𝑛𝑖
𝑁
𝑦𝑖 =
(3.6)
We adopt capital letters for extensive variables and reserve lowercase letters for intensive, that is, specific (per unit mass or mole) variables. The molecular mass, 𝑀, of a
mixture
𝑀=
𝑛
∑
𝑦𝑖 𝑀𝑖
(3.7)
𝑖=1
is the conversion factor required between molar intensive and mass intensive units. For
example, the mass intensive (specific) gas constant 𝑅 is related to the molar intensive
universal gas constant 𝑅u by
𝑅=
𝑅u
𝑀
(3.8)
where 𝑅u = 8.314 kJ/(kmol K).
The internal energy 𝑈 (kJ) of a mixture is the sum of the internal energy 𝑢𝑖 of all 𝑛
components
𝑈=
𝑛
∑
𝑚𝑖 𝑢 𝑖
(3.9)
𝑖=1
The specific internal energy 𝑢 (kJ/kg) is
𝑢=
𝑛
∑
𝑥𝑖 𝑢 𝑖
(3.10)
𝑖=1
The internal energy 𝑈 of a mixture can also be written on a molar basis as
𝑈=
𝑛
∑
𝑛𝑖 𝑢̄ 𝑖
(3.11)
𝑖=1
where the molar intensive properties are denoted with an overbar. The specific molar
internal energy (kJ/kmol) is the mole fraction weighted sum of the component internal
energies:
𝑢̄ =
𝑛
∑
𝑖=1
𝑦𝑖 𝑢̄ 𝑖
(3.12)
68
Fuel, Air, and Combustion Thermodynamics
Analogous relations for the enthalpy 𝐻 (kJ) and specific enthalpy ℎ (kJ/kg) are
𝐻=
𝑛
∑
𝑚𝑖 ℎ 𝑖
(3.13)
𝑥𝑖 ℎ𝑖
(3.14)
𝑖=1
ℎ=
𝑛
∑
𝑖=1
Note that the enthalpy ℎ𝑖 of a component is evaluated at the total pressure 𝑃
ℎ𝑖 = 𝑢𝑖 + 𝑃 𝑣𝑖
On a molar basis, the enthalpy is
𝐻=
𝑛
∑
𝑛𝑖 ℎ̄ 𝑖
(3.15)
𝑦𝑖 ℎ̄ 𝑖
(3.16)
𝑖=1
ℎ̄ =
𝑛
∑
𝑖=1
The enthalpy is defined using a standardized reference state, 𝑇 = 298.15 K and
𝑃o =1 bar. The standardized enthalpy has two parts, the enthalpy related to the chemical bond energy needed to form the substance from its elements, defined as the enthalpy of
formation, ℎ̄ of , and the enthalpy related to the temperature 𝑇 . Values of specific enthalpy
at other states are determined relative to this standardized reference state, as shown by
Equation 3.17.
ℎ̄ 𝑖 (𝑇 ) = ℎ̄ of + (ℎ̄ 𝑖 (𝑇 ) − ℎ̄ of )
(3.17)
Tabular molar specific enthalpy data for elemental gases and combustion products is
given in Appendicies B.3--B.8. The enthalpy of formation ℎ̄ of of the stable form of the
elements such as hydrogen H2 , oxygen O2 , nitrogen N2 , and solid carbon C(s) is assigned
a value of zero at the reference temperature 𝑇 =298.15 K. For compounds, the enthalpy
of formation is the enthalpy required to form the compound from its elements in their
stable state. The enthalpy of formation of CO2 is −393,522 kJ/kmol = −8942 kJ/kg.
Due to the difference in the bond energies, the enthalpy of CO2 (−8942 kJ/kg) at the
standard reference state is consequently less than the enthalpy of its elements C and O2 (0
kJ/kg) at the same reference state. Similarly, the enthalpy of formation of H2 O vapor is
−241,826 kJ/kmol = −13, 424 kJ/kg.
The constant pressure and constant volume specific heats are defined as follows:
( )
𝜕ℎ
(3.18)
𝑐𝑝 =
𝜕𝑇 𝑝
( )
𝜕𝑢
𝑐𝑣 =
(3.19)
𝜕𝑇 𝑣
Useful relationships between ideal gas specific heats 𝑐𝑣 , 𝑐𝑝 and the gas constant 𝑅 can be
developed from the definition of enthalpy ℎ:
ℎ = 𝑢 + 𝑝𝑣
= 𝑢 + 𝑅𝑇
(3.20)
Thermodynamic Properties of Ideal Gas Mixtures
69
in differential form,
𝑑ℎ = 𝑑𝑢 + 𝑅𝑑𝑇
𝑐𝑝 𝑑𝑇 = 𝑐𝑣 𝑑𝑇 + 𝑅𝑑𝑇
(3.21)
so
𝑅 = 𝑐𝑝 − 𝑐𝑣
𝑐𝑣
1
=
𝑅
𝛾 −1
𝑐𝑝
𝛾
=
𝑅
𝛾 −1
(3.22)
The entropy 𝑆 (kJ/K) of a mixture is the sum of the entropy of each component
𝑆=
𝑛
∑
𝑛
∑
𝑚𝑖 𝑠𝑖 =
𝑛𝑖 𝑠̄𝑖
(3.23)
𝑖=1
𝑖=1
From the Gibbs equations, the mass and molar specific entropy, 𝑠𝑖 , (kJ/(kg K)) and 𝑠̄𝑖
(kJ/(kmol K)) of component 𝑖 are
𝑠𝑖 (𝑇 , 𝑃 ) = 𝑠o𝑖 (𝑇 ) − 𝑅𝑖 ln(𝑃𝑖 ∕𝑃o )
(3.24)
𝑠̄𝑖 (𝑇 , 𝑃 ) = 𝑠̄o𝑖 (𝑇 ) − 𝑅u ln(𝑃𝑖 ∕𝑃o )
(3.25)
Note the entropy of a component is evaluated at its partial pressure, 𝑃𝑖 , defined as
(3.26)
𝑃𝑖 = 𝑦𝑖 𝑃
following Dalton’s rule of additive pressures for an ideal gas mixture, in which the partial
pressure is the pressure that a component would exert if it occupied the entire volume at
the given temperature.
A standardized reference state is also defined for entropy. The third law of thermodynamics, postulated by W. Nerst (1864--1941), sets the entropy of pure crystalline elements
and compounds to zero at a temperature of 0 K, which has been chosen as the reference
state for entropy. Values of the specific entropy at other states are determined relative to
0 K. The standard entropy terms 𝑠o𝑖 and 𝑠̄o𝑖 depend only on temperature 𝑇 , and are the mass
and molar specific entropies of a component at the reference pressure 𝑃o , that is, 𝑃𝑖 = 𝑃o =
1 bar. Tabular standard entropy data for ideal gases at 𝑃o =1 bar is given in Appendices
B.3--B.8. For example, at 298 K the molar standard entropy of carbon dioxide, CO2 , is
213.794 kJ/(kmol K).
Substitution of Equation 3.24 into Equation 3.23 yields convenient relations for the
mixture mass (𝑠) and molar (𝑠)
̄ specific entropies with separate pressure and temperature
dependent terms:
𝑠 = −𝑅 ln (𝑃 ∕𝑃o ) +
𝑛
∑
𝑥𝑖 (𝑠o𝑖 − 𝑅𝑖 ln 𝑦𝑖 )
(3.27)
𝑖=1
𝑠̄ = −𝑅u ln (𝑃 ∕𝑃o ) +
𝑛
∑
𝑖=1
𝑦𝑖 (𝑠̄o𝑖 − 𝑅u ln 𝑦𝑖 )
(3.28)
70
Fuel, Air, and Combustion Thermodynamics
The Gibbs free energy 𝐺 of an ideal gas mixture is defined as
𝐺 = 𝐻 − 𝑇𝑆 =
𝑛
∑
𝑛𝑖 𝑔̄𝑖
𝑖=1
=
𝑛
∑
(3.29)
𝑛𝑖 (ℎ̄ 𝑖 − 𝑇 𝑠̄𝑖 )
𝑖=1
EXAMPLE 3.1
Properties of Ideal Gas Mixtures
Compute the molecular mass 𝑀, mass specific enthalpy ℎ (kJ/kg), mass specific entropy
𝑠 (kJ/kg K), and mass specific Gibbs free energy 𝑔 (kJ/kg) of a mixture of combustion
products at 𝑃 =2000 kPa and 𝑇 =1000 K. The constituents and their mole fractions are
Species
CO2
H2 O
N2
CO
H2
y𝑖
0.109
0.121
0.694
0.0283
0.0455
SOLUTION Using the tabular ideal gas data in Appendices B.3--B.8, the following table of
component properties can be generated, and the mixture properties computed:
Species
CO2
H2 O
N2
CO
H2
y𝑖
𝑀𝑖
(kg/kmol)
ℎ̄ of
(kJ/kmol)
ℎ̄ 𝑖 − ℎ̄ of
(kJ/kmol)
𝑠̄o𝑖
(kJ/(kmol K))
𝑦𝑖 (𝑠̄o𝑖 − 𝑅u ln 𝑦𝑖 )
(kJ/(kmol K))
0.109
0.121
0.694
0.0283
0.0455
44.01
18.015
28.013
28.01
2.016
−393, 522
−241, 826
0
−110, 527
0
33,397
26,000
21,463
21,686
20,663
269.30
232.74
228.17
234.54
166.22
31.36
30.28
160.05
7.48
8.73
∑
1. Molecular mass: 𝑀 = 𝑦𝑖 𝑀𝑖 = 27.3 kg/kmol
∑
2. Molar specific enthalpy: ℎ̄ = 𝑦𝑖 ℎ̄ 𝑖 = −52, 047 kJ/kmol
∑
3. Molar specific entropy: 𝑠̄ = −𝑅u ln (𝑃 ∕𝑃o ) + 𝑦𝑖 (𝑠̄o𝑖 − 𝑅u ln 𝑦𝑖 ) = 213.5 kJ/(kmol K)
4. Molar specific Gibbs free energy: 𝑔̄ = ℎ̄ − 𝑇 𝑠̄ = −265, 550 kJ/kmol
Therefore,
̄
ℎ = ℎ∕𝑀
= −1906 kJ/kg
𝑠 = 𝑠∕𝑀
̄
= 7.82 kJ/kg K
𝑔 = 𝑔∕𝑀
̄
= −9727.0 kJ/kg
Comment: The entropy is calculated using the partial pressure, and the values of enthalpy
and Gibbs free energy are relatively low, since this is a mixture of the products of combustion.
Thermodynamic Properties of Ideal Gas Mixtures
71
Specific Heat of Fuel--Air Mixtures
If the composition, that is, mole fractions, of the fuel--air--residual gas mixture are known,
the thermodynamic properties ℎ, 𝑢, 𝑠, and 𝑣 of the mixture are found by application of the
above property relations. The constant pressure specific heat of the fuel--air mixture, 𝑐𝑝 ,
requires a more detailed analysis. The equilibrium constant pressure specific heat depends
not only on the change in enthalpy but also on the change in mixture composition as a
function of temperature. It is defined as
( )
𝜕ℎ
(3.30)
𝑐𝑝 =
𝜕𝑇 𝑝
̄
Since ℎ = ℎ∕𝑀,
𝜕ℎ
1 𝜕 ℎ̄
𝜕 1
=
+ ℎ̄
( )
𝜕𝑇
𝑀 𝜕𝑇
𝜕𝑇 𝑀
Differentiating the molar specific heat with respect to temperature,
ℎ̄ =
𝑛
∑
𝑦𝑖 ℎ̄ 𝑖 ,
(3.31)
(3.32)
𝑖=1
𝑛
𝑛
∑
𝜕 ℎ̄ ∑ ̄ 𝜕𝑦𝑖
𝜕 ℎ̄
𝑦𝑖
=
+
ℎ
𝜕𝑇
𝜕𝑇 𝑖=1 𝑖 𝜕𝑇
𝑖=1
results in
[
1
𝑐𝑝 =
𝑀
𝑛
∑
𝑦𝑖 𝑐𝑝𝑖 +
𝑛
∑
𝑖=1
𝑖=1
ℎ̄ 𝑖
𝜕𝑦𝑖
𝜕𝑇
]
−
ℎ̄ 𝜕𝑀
𝑀 2 𝜕𝑇
∑
The molecular mass is 𝑀 = 𝑦𝑖 𝑀𝑖 , so upon substitution,
[ 𝑛
]
𝑛
𝑛
∑
𝜕𝑦𝑖
𝜕𝑦𝑖
1 ∑
ℎ̄ ∑
̄
𝑀
𝑦𝑐 +
−
ℎ
𝑐𝑝 =
𝑀 𝑖=1 𝑖 𝑝𝑖 𝑖=1 𝑖 𝜕𝑇
𝑀 𝑖=1 𝑖 𝜕𝑇
(3.33)
(3.34)
(3.35)
The frozen specific heat, 𝑐𝑝,𝑓 is computed holding the composition constant. It is
defined as
𝑐𝑝,𝑓 =
𝑛
∑
𝑦𝑖 𝑐𝑝𝑖
(3.36)
𝑖=1
Note the important role that the changes in mole fraction with respect to pressure and
temperature, 𝜕𝑦𝑖 ∕𝜕𝑃 and 𝜕𝑦𝑖 ∕𝜕𝑇 , have in determination of the mixture specific heat.
For computer calculations it is awkward to deal with tabular data. For this reason,
the specific heats of various species have been curve-fitted to polynomials by minimizing
the least-squares error (Gordon and McBride, 1994). The function we will employ for any
given species is
𝑐𝑝
𝑅
=
𝑐̄𝑝
𝑅u
= 𝑎1 + 𝑎2 𝑇 + 𝑎 3 𝑇 2 + 𝑎4 𝑇 3 + 𝑎5 𝑇 4
(3.37)
Since for an ideal gas, 𝑑ℎ = 𝑐𝑝 𝑑𝑇 and 𝑑𝑠 = (𝑐𝑝 ∕𝑇 )𝑑𝑇 , it follows that the enthalpy
and standard entropy at atmospheric pressure are
𝑎
𝑎
𝑎
𝑎
𝑎
ℎ̄
ℎ
=
= 𝑎1 + 2 𝑇 + 3 𝑇 2 + 4 𝑇 3 + 5 𝑇 4 + 6
𝑅𝑇
𝑅u 𝑇
2
3
4
5
𝑇
(3.38)
72
Fuel, Air, and Combustion Thermodynamics
–5000
–6000
Enthalpy (kJ/kg)
–7000
CO2
–8000
–9000
H2O
–10000
–11000
–12000
–13000
Figure 3.1 Enthalpy versus
temperature curve fits for CO2
and H2 O
–14000
300
800
1300
1800
2300
Temperature (K)
𝑎
𝑎
𝑎
𝑠o
𝑠̄o
= 𝑎1 ln 𝑇 + 𝑎2 𝑇 + 3 𝑇 2 + 4 𝑇 3 + 5 𝑇 4 + 𝑎7
=
𝑅
𝑅u
2
3
4
2800
(3.39)
where 𝑎6 and 𝑎7 are constants of integration determined by matching the enthalpy and
entropy to a zero datum at some reference temperature. As discussed above, the reference
temperature for enthalpy is chosen to be 298.15 K with the enthalpy of H2 , O2 , N2 , and
C(s) set to zero.
Values of the curve-fit constants for several species of interest in combustion, CO2 ,
H2 O, N2 , O2 , CO, H2 , H, O, OH, and NO are given in Appendices C.2 and C.3 for
the temperature ranges 300--1000 K and 1000--3000 K. Similar curve-fit coefficients for
several fuels are also given in Appendix C.1.
The mass-specific enthalpies of CO2 and H2 O given by Equation 3.38 are plotted
versus temperature in Figure 3.1. At 298 K, the enthalpy of CO2 is −8942 kJ/kg, and the
enthalpy of H2 O vapor is −13, 424 kJ/kg, consistent with the definition of enthalpy of
formation. Note that the slope of the H2 O curve is steeper than that of the CO2 curve, as a
result of the greater specific heat of the water vapor.
Thermodynamic data for elements, combustion products, and many pollutants are also
available in a compilation published by the National Institute of Standards and Technology
(NIST) called the JANAF Tables (Chase, 1998). For single-component fuels, the data
presented by Stull et al. (1969) is in the same format as that of the JANAF Tables.
For several decades, S. Gordon and B. McBride at the NASA Glenn Research Center
provided least-square coefficients of thermodynamic property data for use in computer
programs. A representative listing of their publications, for example, McBride et al. (1993),
Gordon and McBride (1994), and McBride et al. (2002), is given in the chapter references.
In addition to these references, a compilation by Rossini (1953) is useful for hydrocarbon
fuels at temperatures as high as 1500 K.
3.3 LIQUID--VAPOR--GAS MIXTURES
The thermodynamics involved with fuel injection and vaporization, water injection, and
water condensation can be complicated, as the fuel--air mixtures are composed of more
than one thermodynamic phase. Fortunately, we can make some simplifications that are
quite accurate for our intended use.
Liquid--Vapor--Gas Mixtures
73
First, let us consider a pure substance in terms of its compressed liquid, saturated liquid,
saturated vapor, and superheated vapor states. The simplifications that we will introduce
are
1. Compressed and saturated liquids are incompressible.
2. Saturated and superheated vapors are ideal gases.
For an incompressible substance, it can be shown that the internal energy and entropy
depend only on temperature. Hence the approximation for compressed liquids can be
𝑢 = 𝑢f (𝑇 )
(3.40)
𝑠 = 𝑠f (𝑇 )
(3.41)
where the notation 𝑢f (𝑇 ) and 𝑠f (𝑇 ) denote the internal energy and entropy of saturated
liquid at the temperature 𝑇 . The enthalpy of a compressed liquid depends on pressure, and
it is consistent with Equations 3.40 and 3.41 to assume that
ℎ = ℎof + (ℎ − ℎ298 ) + (𝑃 − 𝑃atm ) 𝑣
(3.42)
where ℎof (𝑇 ) is the enthalpy of formation of the compressed liquid at standard atmospheric
pressure (101.25 kPa) and temperature (298 K). The only property remaining to be prescribed is the specific volume 𝑣. Let us choose it to be the specific volume of compressed
liquid at atmospheric pressure as these data are readily available.
(3.43)
𝑣 = 𝑣o (𝑇1 )
where 𝑇1 is the initial temperature in the process being analyzed.
To introduce the enthalpy of vaporization into our equations of state for the liquids is
convenient, since these data are usually easier to find than the saturated liquid data. We
then have
ℎ = ℎ𝑔 − ℎfg + (𝑃 − 𝑃atm ) 𝑣
(3.44)
Unlike specific volume, data for the enthalpy of vaporization at saturation pressure are
readily available. Hence we choose
(3.45)
ℎfg = ℎfg (𝑇1 )
where, again, 𝑇1 is the temperature at the start of the process being analyzed.
Typically
|𝑃 − 𝑃atm |𝑣 ≪ |ℎfg |
and
|𝑃 𝑣| ≪ |ℎ|
(3.46)
so that in many cases we can write for liquids
ℎ ≈ ℎ𝑔 − ℎfg = ℎof + (ℎ − ℎ298 ) − ℎfg
(3.47)
𝑢≈ℎ
(3.48)
Table 3.1 gives the molar enthalpy of formation for the vapor and liquid state, molar
enthalpy of vaporization, the saturation pressure, and the specific volume of compressed
liquid, for several liquid fuels and water at T = 298 K. The entropy of vaporization is
the difference between the enthalpies of formation of the liquid and the gaseous states.
Table 3.2 gives most of the same information for octane but as a function of temperature.
For example, at 𝑇 = 298 K, the molar enthalpy ℎ̄ of liquid octane is ℎ = ℎ̄ of − ℎ̄ fg =
−208.45 −41.51 = −249.96 kJ/mol.
74
Fuel, Air, and Combustion Thermodynamics
Table 3.1 Ideal Gas Enthalpy of Formation, Enthalpy of Vaporization, Saturation Vapor
Pressure, and Specific Volume of Some Liquid Fuels at 𝑇 = 298 K
Formula
Name
CH3 NO2
CH4 O
C2 H6 O
C4 H10 O
C5 H12
C6 H14
C6 H6
C7 H17
C8 H18
C8 H18
C8 H10
C12 H26
C14 H30
C14.4 H24.9
C16 H34
C19 H40
H2 O
Nitromethane
Methanol
Ethanol
Ethyl ether
Pentane
Hexane
Benzene
Gasoline
Octane
Isooctane
Ethylbenzene
Dodecane
Tetradecane
Diesel fuel
Hexadecane (cetane)
Nonadecane
Water
ℎ̄ of,gas
(MJ/kmol)
ℎ̄ of,liquid
(MJ/kmol)
−74.73
−201.17
−234.81
−252.21
−146.00
−167.19
+82.93
−267.12
−208.45
−224.14
+29.29
−290.87
−332.13
−100.00
−373.34
−435.14
−241.83
−113.1
−239.09
−277.15
−278.74
−172.44
−198.68
+48.91
−305.63
−249.96
−259.25
−12.72
−229.55
−403.37
−174.08
−454.48
−530.17
−285.85
ℎ̄ fg
(MJ/kmol)
𝑃sat
(bar)
𝑣o
(m3 /kg)
38.37
37.92
42.34
26.53
26.44
31.49
34.02
38.51
41.51
35.11
42.01
61.32
71.24
74.08
81.14
95.03
44.02
0.050
0.186
0.084
0.733
0.710
0.242
0.129
0.879 ×10−3
1.264 ×10−3
1.267 ×10−3
1.413 ×10−3
1.597 ×10−3
1.514 ×10−3
1.138 ×10−3
1.449 ×10−3
1.423 ×10−3
1.445 ×10−3
1.153 ×10−3
1.336 ×10−3
1.311 ×10−3
1.176 ×10−3
1.293 ×10−3
1.286 ×10−3
1.000 ×10−3
0.022
0.071
0.013
< 0.001
<0.001
<0.001
<0.001
0.0317
Source: 𝑃sat and 𝑣o are from the CRC Handbook of Chemistry and Physics (2012--2013); ℎ̄ fg is from Vargaflik
(1975); and ℎ̄ of is from Stull et al. (1969).
Table 3.2 Enthalpy of Vaporization, Saturation Vapor Pressure,
and Specific Volume of Octane, C8 H18
T (K)
298
325
350
375
400
425
450
475
500
550
ℎ̄ fg (MJ/kmol)
41.51
39.99
38.22
36.45
34.46
32.35
30.07
27.49
24.32
20.35
𝑃sat (bar)
0.018
0.075
0.211
0.503
1.058
2.003
3.500
5.707
8.837
13.210
𝑣o (m3 /kg)
1.432 ×10−3
1.479 ×10−3
1.527 ×10−3
1.579 ×10−3
1.639 ×10−3
1.708 ×10−3
1.789 ×10−3
1.890 ×10−3
2.022 ×10−3
2.214 ×10−3
Source: Vargaftik (1975).
The saturated vapor pressure is needed for calculations of fuel droplet evaporation. The
Clausius--Clapeyron equation for the saturated vapor pressure as a function of temperature
is
ℎfg
𝑑𝑃sat
(3.49)
=
𝑑𝑇
𝑇 𝑣fg
75
Liquid--Vapor--Gas Mixtures
Table 3.3 Curve-fit Coefficients for Antoine’s Equation for Saturation Vapor Pressure 𝑃sat (bar)
Formula
Name
a
b
c
CH3 NO2
CH4 O
C2 H6 O
C8 H18
C12 H26
C14 H30
C16 H34
H2 O
Nitromethane
Methanol
Ethanol
Octane
Dodecane
Tetradecane
Hexadecane (cetane)
Water
4.1135
5.1585
4.9253
4.0487
4.1055
4.1373
4.1731
5.4022
1229.6
1569.6
1432.5
1355.1
1625.9
1739.6
1845.7
1838.67
−76.221
−34.846
−61.819
−63.633
−92.839
−105.62
−117.05
−31.737
Source: www.webbook.nist.gov
For conditions away from the critical point, we can assume ideal gas behavior 𝑣fg ≃ 𝑣𝑔 ≃
𝑅𝑇 ∕𝑃sat , so Equation 3.49 becomes
𝑑𝑃sat
1 ℎfg (𝑇 )
=
𝑑𝑇
𝑃sat
𝑅 𝑇2
(3.50)
The saturated vapor pressure is typically found from an integrated form of the
Clausius--Clapeyron equation, called Antoine’s equation. Antoine’s Equation 3.51, has
empirical coefficients, 𝑎, 𝑏, and 𝑐 that are curve fits for a given fluid. Table 3.3 lists
the curve fit coefficients for Antoine’s equation for several liquid fuels and water, and
temperature 𝑇 in Kelvin.
]
[
𝑏
(3.51)
log10 (𝑃sat ) = 𝑎 −
𝑇 +𝑐
For example, Equation 3.51 predicts that the saturated vapor pressure 𝑃sat of a cetane
droplet at 𝑇 = 500 K is 0.226 bar.
To integrate the Clausius--Clapeyron equation for the saturation pressure, one needs
to know the temperature dependence of the enthalpy of vaporization, ℎ̄ fg . Equation 3.52 is
a curve fit for the molar enthalpy of vaporization ℎ̄ fg as a function of the temperature ratio
𝑇 ∕𝑇𝑐 , where 𝑇𝑐 is defined as the temperature at the critical point where ℎ̄ fg = 0. Table 3.4
lists the curve-fit coefficients for Equation 3.52 for several liquid fuels.
ℎ̄ fg = 𝐴 𝑒(−𝛼𝑇 ∕𝑇𝑐 ) (1 − 𝑇 ∕𝑇𝑐 )𝛽
(3.52)
Table 3.4 Curve-fit Coefficients for Enthalpy of Vaporization ℎ̄ fg (kJ/mole)
Formula
Name
A
(MJ/kmol)
𝛼
𝛽
𝑇c
(K)
𝑃𝑐
(bar)
CH3 NO2
CH4 O
C2 H6 O
C8 H18
C14 H30
Nitromethane
Methanol
Ethanol
Octane
Tetradecane
53.33
45.30
50.43
58.46
95.66
0.2732
−0.3100
−0.4475
0.1834
0.2965
0.2732
0.4241
0.4989
0.3324
0.2965
588.0
512.6
513.9
568.8
694.0
58.7
81.0
63.0
24.9
14.4
Source: www.webbook.nist.gov
76
Fuel, Air, and Combustion Thermodynamics
We will also deal with mixtures of gases in contact with a liquid phase. For example,
the products of combustion can include water both in the vapor and in the liquid states. In
these cases, we assume the following:
1. The liquid contains no dissolved gases.
2. The gases are ideal.
3. At equilibrium, the partial pressure of the water vapor is equal to the saturation pressure
corresponding to the mixture temperature.
Note that the temperature of exhaust gases is usually high enough, so all of the water is
in vapor form, except during engine warmup conditions. In that case, liquid water can be
seen dripping from vehicle tail pipes, since the exhaust system temperature is below the
dew point temperature.
Let 𝜒 denotes the quality of the condensable substance, that is, the ratio of the vapor
mass or moles to the total liquid and vapor mass or moles. The quality 𝜒 has the same value
on a mole or mass basis.
𝑚H2 O,g
𝑛H O,g
=
(3.53)
𝜒= 2
𝑛H2 O
𝑚H2 O
The enthalpy of the system is then
𝐻 = (1 − 𝜒) 𝑚1 (ℎg,1 − ℎfg ) + 𝜒 𝑚1 ℎg,1 +
𝑛
∑
𝑚𝑖 ℎ 𝑖
(3.54)
𝑖=2
or
ℎ=
𝑛
∑
(3.55)
𝑥𝑖 ℎ𝑖 − (1 − 𝜒)𝑥1 ℎfg
𝑖=1
The indexing is chosen so that 𝑖 = 1 corresponds to the condensable substance and 𝑖 =
2, … , 𝑛 corresponds to all other gases. Likewise, the system volume is
𝑉 = (1 − 𝜒) 𝑚1 (𝑣g,1 − 𝑣fg ) + 𝜒 𝑚1 𝑣g,1 +
𝑛
∑
𝑚𝑖 𝑣𝑖
(3.56)
𝑖=2
and the specific volume is
𝑣=
𝑛
∑
𝑥𝑖 𝑣𝑖 − (1 − 𝜒)𝑥1 𝑣fg
(3.57)
𝑖=1
where the 𝑣𝑖 are computed at the total pressure 𝑃 in accordance with the Amagat--Leduc
Law of additive volumes.
3.4 STOICHIOMETRY
A stoichiometric reaction is defined such that the fuel burns completely and the only products are carbon dioxide and water. This reaction is used as a reference case, and we will
treat a wider variety of combustion products in subsequent sections. The product composition resulting from fuel--air combustion depends on the stoichiometry. For example, if
the reactants are fuel-rich, there is not enough oxygen to react completely with the fuel,
resulting in the formation of additional products such as CO and 𝐻2 , and if the mixture
is fuel-lean, there is not enough fuel to consume the oxygen, so there will be unburned
oxygen in the product mixture.
Stoichiometry
77
Let us represent the chemical formula of a fuel as Ca H𝑏 O𝑐 N𝑑 and assume the following
stoichiometric reaction
Ca H𝑏 O𝑐 N𝑑 + 𝑎s (O2 + 3.76N2 ) → 𝑛1 CO2 + 𝑛2 H2 O + 𝑛3 N2
(3.58)
and solve for 𝑎s , the stoichiometric molar air--fuel ratio, and the moles 𝑛𝑖 (𝑖 = 1, 2, 3) that
describe the product composition. Note that the stoichiometric reaction is expressed per
mole of fuel. We know that atoms are conserved, so using atomic species balance we can
write
C∶
𝑎 = 𝑛1
H∶
𝑏 = 2𝑛2
O∶
𝑐 + 2𝑎s = 2𝑛1 + 𝑛2
N∶
𝑑 + 2 × 3.76 × 𝑎s = 2𝑛3
(3.59)
Solution of these four equations gives
𝑎s = 𝑎 +
𝑏 𝑐
−
4 2
𝑛1 = 𝑎
𝑏
2
)
(
𝑑
𝑏 𝑐
𝑛3 = + 3.76 𝑎 + −
2
4 2
𝑛2 =
(3.60)
The stoichiometric mass air--fuel ratio 𝐴𝐹s can be determined from 𝑎s and the fuel--air
mixture molecular weights:
( )
28.85 (4.76 𝑎s )
𝑚a
=
𝐴𝐹s =
(3.61)
𝑚f s (12.01 𝑎 + 1.008 𝑏 + 16.00 𝑐 + 14.01 𝑑)
A dimensionless measure of the fuel--air ratio is the fuel--air equivalence ratio, 𝜙, which
is defined in Equation 3.62 as the actual fuel--air ratio, 𝐹 𝐴, divided by the stoichiometric
fuel--air ratio, 𝐹 𝐴s ,
𝜙=
𝐹𝐴
𝐹 𝐴s
(3.62)
The reciprocal of 𝜙 is 𝜆, the air--fuel ratio, 𝐴𝐹 , divided by the stoichiometric air--fuel ratio,
𝐴𝐹s :
𝜆=
𝐴𝐹
𝐴𝐹s
(3.63)
The equivalence ratio has the same value on a mole or mass basis. If 𝜙 < 1 the mixture is
lean, if 𝜙 > 1 the mixture is rich, and if 𝜙 = 1 the mixture is stoichiometric.
Typical stoichiometric air--fuel ratio values of various fuels are given in Table 3.4. For
hydrocarbons, that is (𝑐 = 𝑑 = 0), the mass stoichiometric air--fuel ratio 𝐴𝐹s is approximately 15, and not a strong function of the number of carbon atoms. Table 3.4 also lists
the molar stoichiometric air--fuel ratio, 𝑎s , the CO2 , and H2 O product mole fractions, and
the last column in the table is the equilibrium product water quality, 𝜒eq , at a reference
temperature of 298.15 K.
78
Fuel, Air, and Combustion Thermodynamics
Table 3.5 Molecular Mass, Stoichiometric Air--Fuel Ratios, and Product Mole Fractions and
Quality
𝑀
Chemical Formula (kg/kmole)
Fuel
Hydrogen
Methane
Ammonia
Methanol
Propane
Ethanol
Nitromethane
Gasoline
Octane
Diesel
Tetradecane
Hexadecane (Cetane)
H2
CH4
NH3
CH4 O
C3 H8
C2 H6 O
CH3 NO2
C7 H17
C8 H18
C14.4 H24.9
C14 H30
C16 H34
2.02
16.04
17.03
32.04
44.09
46.07
61.04
101.21
114.22
198.04
198.39
226.44
𝐴𝐹s
𝑎s
𝑦CO2
𝑦H2 O
𝜒eq
34.06
17.12
6.05
6.43
15.57
8.94
1.69
15.27
15.03
14.30
14.54
14.56
0.50
2.00
0.75
1.50
5.00
3.00
0.75
11.25
12.50
20.63
21.50
24.5
0.000
0.095
0.000
0.116
0.116
0.123
0.158
0.121
0.125
0.138
0.127
0.128
0.347 0.0061
0.190 0.139
0.311 0.0072
0.231 0.109
0.155 0.178
0.184 0.145
0.237 0.105
0.147 0.201
0.141 0.199
0.119 0.242
0.110 0.207
0.136 0.208
The mole and mass fractions of fuel in a stoichiometric fuel--air mixture are as follows:
𝑛f
1
=
(3.64)
𝑦s =
𝑛f + 𝑛a
1 + 4.76 𝑎s
𝑚f
1
=
(3.65)
𝑥s =
𝑚f + 𝑚a
1 + 𝐴𝐹s
EXAMPLE 3.2
Molecular Mass of Stoichiometric Fuel--Air Mixtures
(a) What is the molecular mass 𝑀mix (kg/kmole) of a stoichiometric mixture of octane
(C8 H18 ) and air? (b) If the mixture fills a automotive engine cylinder with a volume 𝑉 of
9.0 × 10−4 m3 at a pressure 𝑃 = 100 kPa and temperature 𝑇 = 300 K, what is the mass of
the fuel--air mixture in the cylinder?
SOLUTION (a) From Appendix B.1, the molecular mass of air is 𝑀a = 28.97 kg/kmol. From
Table 3.5, the molecular mass 𝑀f of octane is 114.22 kg/kmol, and the stoichiometric
mass air--fuel ratio 𝐴𝐹s is 15.03. Therefore,
𝑦f =
𝑚f ∕𝑀f
𝑛f
=
𝑛f + 𝑛a
𝑚a ∕𝑀a + 𝑚f ∕𝑀f
= … = 0.0166
𝑦a = 1 − 𝑦𝑓 = 0.9834
𝑀mix = 𝑦a 𝑀a + 𝑦𝑓 𝑀f = 30.89 kg/kmol
(b) The mass of the mixture in the cylinder is
𝑚mix =
(100)(9.0 × 10−4 )(30.39)
𝑃𝑉 𝑀
=
= 1.097 g
𝑅u 𝑇
(8.314)(300)
Low-Temperature Combustion Modeling
79
Comment For an automotive class engine, the molecular mass of the fuel--air mixture is
of the order of 30 kg/kmol, and there is typically a gram of combustible mixture in a
cylinder.
3.5 LOW-TEMPERATURE COMBUSTION MODELING
At low temperatures (𝑇 < 1000 K, such as in the product gases in the exhaust stream) and
carbon to oxygen ratios less than one, the overall combustion reaction for any equivalence
ratio can be written as
Ca H𝑏 O𝑐 N𝑑 +
𝑎s
(O + 3.76N2 ) →
𝜙 2
𝑛1 CO2 + 𝑛2 H2 O + 𝑛3 N2 + 𝑛4 O2 + 𝑛5 CO + 𝑛6 H2
(3.66)
This equation assumes the dissociation of reactants is negligible, and a more general case
including dissociation is treated later in this chapter. For reactant C/O ratios greater than
one we would have to add solid carbon C(s) and several other species to the product list,
as we shall see.
For lean (𝜙 < 1) combustion products at low temperature, we will assume no product
CO and H2 , i.e., 𝑛5 = 𝑛6 = 0. In this case, atomic species balance equations are sufficient to determine the product composition, since there are four equations and four
unknowns.
For rich (𝜙 > 1) combustion, we will assume that there is no product O2 , i.e., 𝑛4 = 0.
In this case, there are five unknowns, so we need an additional equation to supplement the
four atom balance equations. Since we have incomplete products of combustion, we need
to assume equilibrium conditions among the product species CO2 , H2 O, CO, and H2 and
no dissociation in order to determine the product composition. This equilibrium reaction is
termed the water--gas shift reaction, given by Equation 3.67:
CO2 + H2 ⇌ CO + H2 O
(3.67)
with the equilibrium constant 𝐾(𝑇 ) for the water--gas shift reaction providing the fifth
equation:
𝐾(𝑇 ) =
𝑛2 𝑛5
𝑛1 𝑛6
(3.68)
The equilibrium constant 𝐾(𝑇 ) equation, Equation 3.69, is a curve fit of JANAF Table
data for 400 < 𝑇 < 3200:
)
(
𝑇
1.761 1.611 0.2803
(3.69)
𝑡=
−
−
ln 𝐾(𝑇 ) = 2.743 −
𝑡
1000
𝑡2
𝑡3
Solutions for both rich and lean cases are given in Table 3.6. In the rich case, the number
of moles of CO, 𝑛5 , is given by the solution of the quadratic equation
𝑛5 =
−𝑏1 +
√
𝑏21 − 4𝑎1 𝑐1
2𝑎1
(3.70)
80
Fuel, Air, and Combustion Thermodynamics
Table 3.6 Low-Temperature (𝑇 < 1000 K) Combustion Products
Species
𝑛𝑖
𝜙≤1
𝜙>1
CO2
H2 O
N2
O2
CO
H2
𝑛1
𝑛2
𝑛3
𝑛4
𝑛5
𝑛6
𝑎
𝑏∕2
𝑑∕2 + 3.76𝑎s ∕𝜙
𝑎s (1∕𝜙 − 1)
0
0
𝑎 − 𝑛5
𝑏∕2 − 𝑑1 + 𝑛5
𝑑∕2 + 3.76𝑎s ∕𝜙
0
𝑛5
𝑑1 − 𝑛 5
where the 𝑎1 , 𝑏1 , and 𝑐1 coefficients are given by
𝑎1 = 1 − 𝐾
𝑏1 = 𝑏∕2 + 𝐾𝑎 − 𝑑1 (1 − 𝐾)
𝑐1 = −𝑎 𝑑1 𝐾
(3.71)
𝑑1 = 2 𝑎s (1 − 1∕𝜙)
EXAMPLE 3.3
Rich Octane Combustion
What are the mole fractions of CO2 , H2 O, CO, N2 , and H2 produced when octane (C8 H18 )
is burned in rich conditions at 𝜙 = 1.2 and 𝑇 = 1000 K?
SOLUTION Since the equivalence ratio of 𝜙 =1.2 is a rich combustion mixture, the product
concentration of O2 is assumed to be zero, that is, 𝑛4 = 0. The combustion equation is
C8 H18 +
𝑎s
(O + 3.76N2 ) →
𝜙 2
𝑛1 CO2 + 𝑛2 H2 O + 𝑛3 N2 + 𝑛5 CO + 𝑛6 H2
The calculation of the product mole fractions proceeds as follows:
𝑎 = 8, 𝑏 = 18, 𝑐 = 𝑑 = 0
𝑎s = 𝑎 + 𝑏∕4 − 𝑐∕2 = 12.5
𝑑1 = 2 𝑎s (1 − 1∕𝜙) = 4.167
𝑡 = 𝑇 ∕1000 = 1
ln 𝐾 = 2.743 − 1.761∕𝑡 − 1.611∕𝑡2 + 0.2803∕𝑡3 = −0.34
𝑎1 = 1 − 𝐾 = 0.295
𝑏1 = 𝑏∕2 + 𝑎𝐾 − 𝑑1 (1 − 𝐾) = 13.41
𝑐1 = −𝑎𝑑1 𝐾 = −23.50
(
)
√
𝑛5 = −𝑏1 + 𝑏21 − 4𝑎1 𝑐1 ∕2𝑎1 = 1.690
𝑛1 = 𝑎 − 𝑛5 = 6.310
(3.72)
Low-Temperature Combustion Modeling
81
𝑛2 = 𝑏∕2 − 𝑑 + 𝑛5 = 6.523
𝑛3 = 𝑑∕2 + 3.76 𝑎s ∕𝜙 = 39.167
𝑛6 = 𝑑1 − 𝑛5 = 2.477
∑
𝑁=
𝑛𝑖 = 56.167
The combustion equation is therefore
C8 H18 + 10.42(O2 + 3.76N2 ) →
6.310CO2 + 6.523H2 O + 39.167N2 + 1.690CO + 2.477H2
so
𝑦CO2 = 𝑛1 ∕𝑁 = 0.112
𝑦H2 O = 𝑛2 ∕𝑁 = 0.116
𝑦N2 = 𝑛3 ∕𝑁 = 0.697
𝑦CO = 𝑛5 ∕𝑁 = 0.0301
𝑦H2 = 𝑛6 ∕𝑁 = 0.0441
Fuel--Air--Residual Gas
In reciprocating engines there is residual gas mixed with the fuel and air, since not all
of the combustion gases leave the cylinder. We need to determine the composition of the
fuel--air--residual gas mixture for analysis of the compression stroke and later for analysis
of the unburned mixture ahead of the flame. The residual gas is assumed to be at a low
enough temperature (𝑇 < 1000 K) so that the species relations in Table 3.6 specify its
composition.
The fuel--air--residual gas mixture will contain both reactants and products. Let us
rewrite the combustion equation as
𝑛′0 Ca H𝑏 O𝑐 N𝑑 + 𝑛′4 O2 + 𝑛′3 N2 →
𝑛′′
CO2 + 𝑛′′
H O + 𝑛′′
N + 𝑛′′
O + 𝑛′′
CO + 𝑛′′
H
1
2 2
3 2
4 2
5
6 2
(3.73)
where
𝑛′𝑖 = reactant coefficient for species 𝑖
𝑛′′
𝑖 = product coefficient for species 𝑖
Adopting similar notation for other symbols, we can develop relations for the species
mass and mole fractions for a mixture of residual gas (r) with residual fraction 𝑓 and
premixed fuel--air (fa).
82
Fuel, Air, and Combustion Thermodynamics
The residual mole fraction 𝑦𝑟 is
𝑦𝑟 =
𝑛r
=
𝑛fa + 𝑛r
𝑛fa
𝑛r
1
+1
Since 𝑓 = 𝑚r ∕𝑚, we can write
𝑚fa
1
= −1
𝑚r
𝑓
and
𝑛fa
𝑚 𝑀
𝑚 𝑀 ′′
= fa r = fa
𝑛r
𝑚r 𝑀fa
𝑚r 𝑀 ′
Upon substitution, the residual mole fraction, 𝑦r , is
[
]−1
𝑀 ′′ 1
(
−
1)
𝑦r = 1 +
𝑀′ 𝑓
(3.74)
The species mole fractions 𝑦𝑖 are
𝑦𝑖 =
𝑛𝑖
𝑛
𝑛
= 𝑖 𝑦fa + 𝑖 𝑦r
𝑁
𝑛fa
𝑛r
Since
𝑦fa = 1 − 𝑦r
𝑦′𝑖 =
𝑛𝑖
𝑛fa
and
𝑦′′
𝑖 =
𝑛𝑖
𝑛r
The species mole fractions are
𝑦𝑖 = (1 − 𝑦r ) 𝑦′𝑖 + 𝑦r 𝑦′′
𝑖
𝑖 = 0, 6
(3.75)
𝑖 = 0, 6
(3.76)
Similarly, the species mass fractions are
𝑥𝑖 = (1 − 𝑓 ) 𝑥′𝑖 + 𝑓 𝑥′′
𝑖
For a constant composition mixture, the equilibrium specific heat is identical to the
frozen specific heat. However, for a reacting fuel, air, and residual gas mixture, the partial
derivatives of the constituent mole fractions 𝑦𝑖 with respect to temperature are required for
determination of the equilibrium specific heat. Since neither the residual mole fraction 𝑦r
nor the reactant mole fractions 𝑦′𝑖 depend on temperature, differentiation of Equation 3.75
with respect to temperature yields
𝜕𝑦′′
𝜕𝑦𝑖
1 𝜕𝑛𝑖
= 𝑦r 𝑖 = 𝑦 r
𝜕𝑇
𝜕𝑇
𝑁 𝜕𝑇
(3.77)
Inspection of Table 3.6 indicates that for lean combustion, none of these six partial
derivatives depend on temperature. For rich combustion, only those derivatives with 𝑛5
(CO) depend on 𝑇 , since the temperature-dependent equilibrium constant 𝐾(𝑇 ) appears in
Low-Temperature Combustion Modeling
83
the solution of the quadratic equation 3.70. Hence, we can write
𝜕𝑦𝑖
=0
𝜕𝑇
(lean)
𝜕𝑦𝑖
1 𝜕𝑛𝑖 𝜕𝑛5 𝜕𝐾
= 𝑦r
𝜕𝑇
𝑁 𝜕𝑛5 𝜕𝐾 𝜕𝑇
(3.78)
(rich)
The terms 𝜕𝑛𝑖 ∕𝜕𝑛5 in Equation 3.78 can be determined by differentiating the 𝜙 > 1 column
in Table 3.6 with respect to 𝑛5 , resulting in: 𝜕𝑛1 ∕𝜕𝑛5 = −1, 𝜕𝑛2 ∕𝜕𝑛5 = 1, 𝜕𝑛3 ∕𝜕𝑛5 = 0,
𝜕𝑛4 ∕𝜕𝑛5 = 0, 𝜕𝑛5 ∕𝜕𝑛5 = 1, and 𝜕𝑛6 ∕𝜕𝑛5 = −1. Also, by differentiating Equation 3.69 with
respect to 𝑇 :
𝜕𝐾 𝜕𝑡
𝜕𝐾
=
𝜕𝑇
𝜕𝑡 𝜕𝑇
(
)
1.761 3.222 0.8409
𝐾
=
+
−
2
3
4
1000
𝑡
𝑡
𝑡
(3.79)
𝜕𝑛5
(𝛼 − 𝑛5 )[𝑛5 + 2𝑎s (1∕𝜙 − 1)]
=−
𝜕𝐾
𝛽∕2 + 𝑛5 + 2𝑎s (1∕𝜙 − 1)
(3.80)
and finally,
The above equations are solved numerically in the Fuel--Air--Residual Gas program
farg.m, which is listed in Appendix F.8. For temperatures between 300 and 1000 K,
the program computes the properties of a fuel--air--residual gas mixture given the mixture
pressure, temperature, the fuel--air equivalence ratio, and the residual mass fraction. With
the mixture mole fraction composition known, the program then proceeds to compute
the thermodynamic properties of the mixture: enthalpy, entropy, specific volume, internal
energy, and equilibrium specific heat for the given conditions.
Five representative fuels are used in the program, and are identified with the following
fuel id, from 1--5:
1. Methane CH4
2.
3.
4.
5.
Gasoline C7 H17
Diesel C14.4 H24.9
Methanol CH3 OH
Nitromethane CH3 NO2
EXAMPLE 3.4
Fuel--Air--Residual Gas
What are the mole fractions and thermodynamic properties of a fuel--air--residual mixture
of methane (fuel id = 1) at a temperature 𝑇 of 500 K, pressure 𝑃 of 100 kPa, a fuel--air
equivalence ratio 𝜙 of 0.8, and residual fraction 𝑓 of 0.1?
SOLUTION The computation is performed using the Fuel--Air--Residual Gas program RunFarg.m input--output program with inputs 𝑇 , 𝑃 , 𝜙, fuel id, and 𝑓 . As detailed in Appendix F.7, the program RunFarg.m calls the function farg.m, which in turn calls the
function fuel.m for fuel properties, computes residual gas composition according to Table 3.6, computes residual mole fractions and molecular mass of the residual gas, then
computes and outputs the fuel--air--residual gas mixture mole fractions, property values,
and outputs the mole fractions and mixture properties.
84
Fuel, Air, and Combustion Thermodynamics
Input parameters to farg:
T = 500;
% temperature (K)
P = 100;
% pressure (kPa)
phi = 0.8;
% equivalence ratio
f = 0.1;
% residual fraction
fuel_id = 1;
% methane identifer
% call function farg
function [y,h,u,s,v,R, Cp,MW,dvdt,dvdp] =
farg(T, P, phi, fuel_id);
...
The resulting mole fractions and properties are listed below.
Fuel Air Residual Gas Output
Mole Fractions
CO2 =
H2O =
0.0078
0.0155
N2 =
0.7287
O2 =
0.1783
CO =
0.0000
H2 =
H =
0.0000
0.0000
O = 0.0000
OH = 0.0000
NO = 0.0000
Mixture Properties
h(kJ/kg) = -211.0
u(kJ/kg) = -360.3
s (kJ/Kg K) = 7.766
v (m3/kg) = 1.492
Cp (kJ/Kg K) = 1.122
Molecular Mass = 27.86
dvdt = 2.98e-03
dvdp = -1.49e-02
3.6 GENERAL CHEMICAL EQUILIBRIUM
In general, we often consider a combustion problem that has many product species. The fuel
is initially mixed with air with an equivalence ratio 𝜙. After combustion, the products of
reaction are assumed to be in equilibrium at temperature 𝑇 and pressure 𝑃 . The composition
and thermodynamic properties of the product mixture are to be determined. The overall
General Chemical Equilibrium
85
combustion reaction per mole of fuel is
𝑎
Ca H𝑏 O𝑐 N𝑑 + s (O2 + 3.76N2 ) →
𝜙
𝑛1 CO2 + 𝑛2 H2 O + 𝑛3 N2 + 𝑛4 O2 + 𝑛5 CO + 𝑛6 H2
+𝑛7 H + 𝑛8 O + 𝑛9 OH + 𝑛10 NO + 𝑛11 N + 𝑛12 C(s)
(3.81)
+𝑛13 NO2 + 𝑛14 CH4 + …
The condition for equilibrium is usually stated in terms of thermodynamic functions
such as the minimization of the Gibbs or Helmholtz free energy or the maximization of
entropy. If temperature and pressure are used to specify a thermodynamic state, the Gibbs
free energy is most easily minimized, since temperature and pressure are its fundamental
variables. For a product mixture of 𝑛 species, the Gibbs free energy 𝐺 is
𝐺=
𝑛
∑
(3.82)
𝑛𝑗 𝜇𝑗
𝑗=1
The chemical potential, 𝜇𝑗 , of species 𝑗 represents the partial molal Gibbs free energy,
i. e., the partial derivative of the Gibbs free energy with respect to the number of moles of
component 𝑗 holding 𝑇 , 𝑃 and the number of moles of the other components constant.
(
)
𝜕𝐺
(3.83)
𝜇𝑗 =
𝜕𝑛𝑗 T,P,n
𝑖≠𝑗
The equilibrium state can be determined by a Lagrangian multiplier approach, that is,
minimizing the Gibbs free energy subject to constraints. In this case, the constraint is the
conservation of the number of atoms of each reacting species, 𝑏′𝑖 :
𝑏′𝑖 =
𝑛
∑
(3.84)
𝑎𝑖𝑗 𝑛𝑗
𝑗=1
or
𝑏𝑖 − 𝑏′𝑖 = 0
(3.85)
where the index 𝑖 = 1, … 𝑙, the integer 𝑙 is the number of atom types, 𝑎𝑖𝑗 is the number of
atoms of element 𝑖 in species 𝑗, 𝑏′𝑖 is the number of atoms of element 𝑖 in the reactants, and
𝑏𝑖 is the number of atoms of element 𝑖 in the products.
𝑏𝑖 =
𝑛
∑
(3.86)
𝑎𝑖𝑗 𝑛𝑗
𝑗=1
is the number of atoms of element 𝑖 in the products.
Using the Lagrangian optimization procedure, we first define the parameter 𝐵:
𝐵 =𝐺+
𝑙
∑
𝜆𝑖 (𝑏𝑖 − 𝑏′𝑖 )
(3.87)
𝑖=1
where the 𝜆𝑖 are the Lagrangian multipliers, one for each element. The variational condition,
𝛿𝐵 = 0, for equilibrium is
)
(
𝑙
𝑙
𝑛
∑
∑
∑
(𝑏𝑖 − 𝑏′𝑖 )𝛿𝜆𝑖 = 0
(3.88)
𝜆𝑖 𝑎𝑖𝑗 𝛿𝑛𝑗 +
𝜇𝑗 +
𝛿𝐵 =
𝑗=1
𝑖=1
𝑖=1
86
Fuel, Air, and Combustion Thermodynamics
Treating the variations 𝛿𝑛𝑗 and 𝛿𝜆𝑖 as independent
𝜇𝑗 +
𝑙
∑
𝜆𝑖 𝑎𝑖𝑗 = 0
(3.89)
𝑗 = 1, … , 𝑛
𝑖=1
For ideal gases the chemical potential 𝜇𝑗 is
𝜇𝑗 = 𝜇𝑗o + 𝑅u 𝑇 ln (𝑛𝑗 ∕𝑁) + 𝑅u 𝑇 ln (𝑃 ∕𝑃o )
(3.90)
so that
𝜇𝑗o
𝑅u 𝑇
+ ln (𝑛𝑗 ∕𝑁) + ln (𝑃 ∕𝑃o ) +
𝑙
∑
𝜋𝑖 𝑎𝑖𝑗 = 0
𝑗 = 1, … , 𝑛
(3.91)
𝑖=1
where the dimensionless Lagrange multiplier, 𝜋𝑖 , is
(3.92)
𝜋𝑖 = 𝜆𝑖 ∕𝑅u 𝑇
To determine the equilibrium composition using the Lagrange multiplier approach, we
have to solve a set of 𝑛 + 𝑙 + 1 equations. For a given temperature and pressure (𝑇 , 𝑃 ),
Equation 3.91 is a set of 𝑛 equations for the 𝑛 unknowns 𝑛𝑗 , 𝑙 unknowns 𝜋𝑖 , and 𝑁. Equation
3.85 provides an additional 𝑙 equation and we close the set with
𝑁=
𝑛
∑
(3.93)
𝑛𝑗
𝑗=1
Once the composition of the products has been determined, we can now compute
the thermodynamic properties of the equilibrium mixture. Recall that any two of the
independent properties 𝑇 , 𝑃 , 𝐻, 𝑆, 𝑈 , and 𝑉 specify the thermodynamic state. For example,
for constant pressure combustion, the enthalpy is known instead of the temperature. For
this case, we include an equation for the known enthalpy to our set of equations,
𝐻=
𝑛
∑
𝑛𝑗 ℎ̄ 𝑗
(3.94)
𝑗=1
For an isentropic compression or expansion, or expansion to a specified pressure, the
entropy is given instead of enthalpy or temperature. In this case, we have
𝑆=
𝑛
∑
𝑛𝑗 (𝑠̄o𝑗 − 𝑅u ln (𝑛𝑗 ∕𝑁) − 𝑅u ln (𝑃 ∕𝑃o ))
(3.95)
𝑗=1
Finally, if in any case specific volume rather than pressure is known, then we have to
minimize the Helmholtz free energy. In this case, a similar analysis (Gordon and McBride,
1994) shows that Equation 3.91 is replaced by
𝜇𝑗o
𝑅u 𝑇
+ ln (𝑛𝑗 ∕𝑁) + ln (𝑅𝑇 ∕𝑃o 𝑣) +
𝑙
∑
𝜋𝑖 𝑎𝑖𝑗 = 0
𝑗 = 1, … , 𝑛
(3.96)
𝑖=1
For constant volume combustion, the internal energy is known, so we include
𝑈=
𝑛
∑
𝑛𝑗 (ℎ̄ 𝑗 − 𝑅u 𝑇 )
(3.97)
𝑗=1
For an isentropic expansion or compression to a specified volume 𝑣, we include
𝑆=
𝑛
∑
𝑗=1
𝑛𝑗 (𝑠̄o𝑗 − 𝑅u ln (𝑛𝑗 ∕𝑁) − 𝑅u ln (𝑅𝑇 ∕𝑃o 𝑣))
(3.98)
General Chemical Equilibrium
87
Solution of these problems for practical application requires numerical iteration on
a computer. Fortunately, there are now several computer programs available. Thermodynamic properties and equilibrium compositions can be computed using a classic NASA program called CEA (Chemical Equilibrium with Applications), Gordon and Mcbride (1994).
This program uses the minimization of Gibbs free energy approach for computation of the
equilibrium composition of reacting species.
Results illustrating composition shifts with temperature and equivalence ratio are given
in Figures 3.2 and 3.3 for the combustion of C8 H18 at 𝑃 = 50 bar. Composition as a function
of temperature is shown in Figure 3.2. The largest mole fractions are N2 , H2 O, and CO2 .
At this pressure, the composition predicted using Table 3.4 is a good approximation for
100
9
8
7
6
5
= 0.8
N2
= 1.0
N2
= 1.2
N2
4
P = 50 bar
3
Mole fraction
2
10 –1
9
8
7
6
5
H2O
CO
H2O
H2 O
CO2
CO
CO
CO2
O2
NO
OH
4
3
CO2
H2
OH
NO
O2
H2
2
OH
NO
H
O
H
O
H2
10 –2
9
8
7
6
5
H
O2
O
4
3
2
CH4
10 –3
1000
3000
1000
3000
Temperature (K)
1000
3000
Figure 3.2 Equilibrium composition of octane (C8 H18 )--air mixtures for different temperatures at
𝜙 = 0.8, 1.0, and 1.2.
88
Fuel, Air, and Combustion Thermodynamics
Figure 3.3 Equilibrium composition of octane (C8 H18 )--air mixtures as a function of 𝜙 at 𝑇 = 3000
K, and 𝑃 = 50 bar.
temperatures less than about 2000 K. At lower pressures, dissociation is even greater, so
that at atmospheric pressure, Table 3.6 is valid for temperatures less than about 1500 K. As
the reaction temperature is increased above 1500 K, there is an exponential rise in product
species such as CO, NO, OH, O2 , O, H2 , and H. For lean (𝜙 < 1) conditions, the O2
fraction is relatively insensitive to temperature. For rich conditions, the H2 mole fraction
first decreases, then increases with increasing temperature.
Notice that at high temperatures there is a significant amount of nitric oxide (NO).
If any gas in an engine cylinder is raised to these high temperatures, that gas will tend
toward equilibrium at a rate determined by chemical kinetics. Since the chemistry for most
Chemical Equilibrium using Equilibrium Constants
89
species that contribute to the thermodynamic properties is fast enough relative to engine
time scales, in many cases local equilibrium may be assumed. Nitric oxide, however, is
significant even though its concentrations are relatively low because it is an air pollutant.
Unlike the species of thermodynamic importance, its chemistry is not fast enough to assume
that it is in equilibrium concentrations. Likewise, once formed, its concentration ‘‘freezes’’
during the expansion stroke so that even in the low temperature exhaust gases nitric oxides
are found. This will be discussed more fully when we deal with emissions.
Composition as a function of equivalence ratio is illustrated in Figure 3.3. The mole
fraction behavior relative to equivalence ratio is complex. The results show the general
trends expected from Figure 3.4 and Table 3.6. The product species CO and H2 generally
increase with equivalence ratio, while the O2 , NO, OH, and O mole fractions decrease.
If the equivalence ratio 𝜙 is greater than about 4, the product species list becomes
quite large and includes solid carbon, C(s); hydrogen cyanide, HCN; acetylene, C2 H2 ; and
methane, CH4 . Thus, if anywhere in the cylinder there are fuel air pockets where 𝜙 > 3,
such as in diesel or stratified charge engines, there will be a tendency for these species
to form. Similar to nitric oxides, their concentration may freeze when mixed with leaner
pockets or when the temperature drops, so these species can appear in the exhaust. For
example, with diesel engines, even though the engine is running lean, the maximum power
is limited by the appearance of solid carbon (smoke and soot) in the exhaust.
3.7 CHEMICAL EQUILIBRIUM USING EQUILIBRIUM CONSTANTS
This section presents a numerical solution for the properties of equilibrium combustion
products based on an equilibrium constant method applied by Olikara and Borman (1975)
to the gas phase products of combustion of hydrocarbon fuels. The use of equilibrium
constants is also based on the minimization of the Gibbs free energy of the gas mixture;
however, it is algebraically less complex than the Lagrange multiplier approach when
considering restricted species lists. The equilibrium constant method does require however,
that equilibrium reactions, such the water--gas reaction given by Equation 3.67, be specified.
A more complete reaction calculation, such as done in the previous section, needs to
be performed first to determine the significant product species to include in the equilibrium
constant analysis. Inspection of Figures 3.2 and 3.3 shows that if 𝜙 < 3, the only product
species of importance resulting from dissociation are O, H, OH, and NO. Therefore, the
species list in Equation 3.81 can be terminated at 𝑖 = 10; that is, we need to consider only
10 species. Therefore, let us consider the following reaction:
Ca H𝑏 O𝑐 N𝑑 +
𝑎s
(O + 3.76N2 ) →
𝜙 2
𝑛1 CO2 + 𝑛2 H2 O + 𝑛3 N2 + 𝑛4 O2 + 𝑛5 𝐶𝑂 + 𝑛6 H2
(3.99)
+𝑛7 H + 𝑛8 O + 𝑛9 OH + 𝑛10 NO
Atom balancing yields the following four equations:
C∶
𝑎 = (𝑦1 + 𝑦5 ) 𝑁
H∶
𝑏 = (2𝑦2 + 2𝑦6 + 𝑦7 + 𝑦9 ) 𝑁
O∶
𝑐 + 2 𝑎s ∕ 𝜙 = (2𝑦1 + 𝑦2 + 2𝑦4 + 𝑦5 + 𝑦8 + 𝑦9 + 𝑦10 ) 𝑁
N∶
𝑑 + 7.52 𝑎s ∕ 𝜙 = (2𝑦3 + 𝑦10 ) 𝑁
(3.100)
90
Fuel, Air, and Combustion Thermodynamics
where 𝑁 is the total number of moles. By definition, the mole fractions sum to 1:
10
∑
(3.101)
𝑦𝑖 = 1
𝑖=1
From these equations, three constants are defined:
𝑏
𝑎
𝑎
𝑐
𝑑2 = + 2 s
𝑎
𝜙𝑎
𝑑1 =
𝑑3 =
𝑑 7.52𝑎s
+
𝑎
𝜙𝑎
Upon substitution into the atom balance equations, and with some rearrangement,
2𝑦2 + 2𝑦6 + 𝑦7 + 𝑦9 + 𝑑1 𝑦1 − 𝑑1 𝑦5 = 0
2𝑦1 + 𝑦2 + 2𝑦4 + 𝑦5 + 𝑦8 + 𝑦9 + 𝑦10 − 𝑑2 𝑦1 − 𝑑2 𝑦5 = 0
2𝑦3 + 𝑦10 − 𝑑3 𝑦1 − 𝑑3 𝑦5 = 0
∑
𝑦𝑖 = 1
(3.102)
We now introduce six gas-phase equilibrium reactions. These reactions include the
dissociation of hydrogen, oxygen, water, and carbon dioxide, and the formation of OH and
NO:
1
H
2 2
⇌H
1
O
2 2
⇌O
𝐾1 =
𝐾2 =
𝑦7 𝑃 1∕2
1∕2
𝑦6
𝑦8 𝑃 1∕2
1∕2
𝑦4
𝑦9
1∕2 1∕2
𝑦4 𝑦6
1
H
2 2
+ 12 O2 ⇌ OH
𝐾3 =
1
O
2 2
+ 12 N2 ⇌ NO
𝐾4 =
H2 + 21 O2 ⇌ H2 O
𝐾5 =
𝑦2
1∕2
𝑦4 𝑦6 𝑃 1∕2
CO + 12 O2 ⇌ CO2
𝐾6 =
𝑦1
1∕2
𝑦4 𝑦5 𝑃 1∕2
𝑦10
(3.103)
1∕2 1∕2
𝑦4 𝑦3
The unit of pressure in the above six equations is in units of atmospheres (atm). Note that
the water--gas shift reaction, given by Equation 3.67, is represented by the last two reaction
equations for 𝐾5 and 𝐾6 . Olikara and Borman (1975) have curve fitted the equilibrium
constants 𝐾𝑖 (𝑇 ) to JANAF Table data for the temperature range 600 < 𝑇 > 4000 K. Their
expressions are of the form
log10 𝐾𝑖 (𝑇 ) = 𝐴𝑖 ln(𝑇 ∕1000) +
𝐵𝑖
+ 𝐶𝑖 + 𝐷𝑖 𝑇 + 𝐸𝑖 𝑇 2
𝑇
(3.104)
Chemical Equilibrium using Equilibrium Constants
Table 3.7
91
Equilibrium Constant 𝐾𝑖 Curve-Fit Coefficients
𝐾𝑖
𝐴𝑖
𝐵𝑖
𝐶𝑖
𝐷𝑖
𝐸𝑖
𝐾1
𝐾2
𝐾3
𝐾4
𝐾5
𝐾6
+0.432168E + 00
+0.310805E + 00
−0.141784E + 00
+0.150879E − 01
−0.752364E + 00
−0.415302E − 02
−0.112464E + 05
−0.129540E + 05
−0.213308E + 04
−0.470959E + 04
+0.124210E + 05
+0.148627E + 05
+0.267269E + 01
+0.321779E + 01
+0.853461E + 00
+0.646096E + 00
−0.260286E + 01
−0.475746E + 01
−0.745744E − 04
−0.738336E − 04
+0.355015E − 04
+0.272805E − 05
+0.259556E − 03
+0.124699E − 03
+0.242484E − 08
+0.344645E − 08
−0.310227E − 08
−0.154444E − 08
−0.162687E − 07
−0.900227E − 08
where 𝑇 is in Kelvin. The equilibrium constant 𝐾𝑖 curve-fit coefficients are listed in
Table 3.7. Given pressure 𝑃 , temperature 𝑇 , and equivalence ratio 𝜙, Equations 3.100,
3.101, and 3.103 will yield eleven equations for the eleven unknowns: the ten unknown
mole fractions 𝑦𝑖 and the unknown total product moles 𝑁.
Substitution of the six individual equilibrium reaction equations into the atom balance
equations results in four equations in four unknowns (𝑦3 , 𝑦4 , 𝑦5 , 𝑦6 ). These four equations
are solved numerically in the Equilibrium Combustion Solver program ecp.m, which is
listed in Appendix F.11. The program computes the product mole fractions and properties
for five representative fuels given the mixture pressure, temperature, and the fuel--air
equivalence ratio. The fuels are identified with the following fuel id, from 1--5:
1. Methane CH4
2.
3.
4.
5.
Gasoline C7 H17
Diesel C14.4 H24.9
Methanol CH3 OH
Nitromethane CH3 NO2
With the mixture mole fraction composition known, one can then proceed to compute
the thermodynamic properties of interest: enthalpy, entropy, specific volume, internal
energy, and specific heat for the given conditions. A reacting mixture of ideal gases
has an enthalpy dependent on temperature and pressure, and computing the mixture
equilibrium specific heat 𝑐𝑝 requires the change in mole fraction due to a change in temperature, so as discussed earlier in this chapter, the mole fraction partial derivatives 𝜕𝑦𝑖 ∕𝜕𝑃
and 𝜕𝑦𝑖 ∕𝜕𝑇 are also computed. The use of the program is detailed in the following example.
EXAMPLE 3.5
Equilibrium Combustion Mole Fraction
What are the mole fractions and mixture properties resulting from the combustion of a
gasoline (fuel id = 2) mixture at a temperature 𝑇 = 3000 K, pressure 𝑃 = 5000 kPa, and a
fuel--air equivalence ratio 𝜙 = 0.8?
SOLUTION The computation is performed using the Equilibrium Combustion Solver input--output
program RunEcp.m with inputs 𝑇 , 𝑃 , 𝜙, and fuel id. As detailed in Appendix F.10,
the program RunEcp.m calls the function ecp.m, which in turn calls the functions
fuel.m for fuel properties and farg.m for initial guess values of mixture properties, then
iterates for converged property values, and finally outputs the mole fractions and mixture
properties.
92
Fuel, Air, and Combustion Thermodynamics
Program RunEcp.m
Input parameters to ecp
T = 3000;
% temperature (K)
P = 5000;
% pressure (kPa)
phi = 0.8;
% equivalence ratio
fuel_id = 2; % gasoline fuel identifer
% call function ecp
function [ierr,y,h,u,s,v,r, cp,mw,dvdt,dvdp] =
ecp(T, P, phi, fuel_id);
...
The resulting mole fractions and properties are listed below. Note that the equilibrium
mole fractions calculated with the equilibrium constant model compare well with the mole
fractions of Figure 3.2 computed by the more general Lagrange multiplier method.
Equilibrium Combustion Solver Output
Mole Fractions
CO2 =
0.0775
H2O = 0.1064
N2 = 0.7203
O2 = 0.0359
CO = 0.0191
H2 = 0.0036
H = 0.0013
O = 0.0030
OH = 0.0133
NO = 0.0196
Mixture Properties
h(kJ/kg) = 1640.4
u(kJ/kg) = 751.5
s (kJ/Kg K) = 8.941
v (m3/kg) = 0.178
cp (kJ/Kg K) = 2.511
Molecular Mass = 28.06
dvdt = 6.81e-05
dvdp = -3.56e-05
The Equilibrium Combustion Solver program can be used to compute general trends
for fuel--air combustion that are not immediately obvious. The effect of temperature on
enthalpy of the combustion products for three different equivalence ratios is shown in Figure 3.4 for the combustion of gasoline at a pressure of 101.3 kPa. Note that the lowest value
of enthalpy occurs at a stoichiometric equivalence ratio, and as the equivalence ratio is
made lean or rich, below 2500 K, the enthalpy increases. The Equilibrium Combustion
Chemical Equilibrium using Equilibrium Constants
93
1500
P = 101.3 kPa
Enthalpy (kJ/kg)
1000
500
0
= 1.2
0.8
–500
1.0
–1000
Figure 3.4 Enthalpy of combustion
products for a gasoline--air
equilibrium mixture for different
temperatures at 𝑃 = 101.3 kPa.
–1500
1500
2000
2500
Temperature (K)
3000
Solver program has been extended to include a wider variety of fuels, Buttsworth
(2002).
This behavior is also shown in Figure 3.5, a plot of the enthalpy of the combustion
products of methanol versus equivalence ratio at pressures of 101 kPa and 2000 kPa. The
enthalpy is a minimum at near stoichiometric conditions, as on either side of stoichiometric,
the combustion is incomplete. If the mixture is lean, there is an excess of unburnt oxygen.
If the mixture is rich, there will be unburnt carbon monoxide. A minimal value of enthalpy
implies that the specific heat of the combustion products is also a minimum, which will
maximize the adiabatic flame temperature, discussed in Chapter 4.
–600
–700
T = 2000 K
Enthalpy (kJ/kg)
–800
–900
–1000
2000
–1100
–1200
Figure 3.5 Enthalpy of
combustion products of a
methanol--air equilibrium
mixture for different 𝜙 at
𝑇 = 2000 K.
–1300
P(kPa) = 101
–1400
0.8
0.9
1
1.1
Equivalence ratio
1.2
1.3
1.4
94
Fuel, Air, and Combustion Thermodynamics
3.8 REFERENCES
BUTTSWORTH, D. (2002), ‘‘Spark Ignition Internal Combustion Engine Modeling using Matlab,’’
Report TR-2002-2, Univ. Southern Queensland, Toowoomba, Australia.
CHASE, M. (1998), NIST - JANAF Thermochemical Tables, 4th edition, J. Chemical and Physical
Reference Data, Monograph No. 9, NIST, Gaithersburg, Maryland, web: http://www.kinetics
.nist.gov/.
CRC Handbook of Chemistry and Physics (2012--2013), 93th ed., CRC Press, Cleveland, Ohio, web:
http://www.hbcpnetbase.com/.
GORDON, S. and B.J. MCBRIDE (1994), ‘‘Computer Program for Calculation of Complex Chemical
Equilibrium Composition, and Applications,’’ NASA RP-1311.
MCBRIDE B., S. GORDON, and M.A. RENO (1993), ‘‘Coefficients for Calculating Thermodynamic and
Transport Properties of Individual Species,’’ NASA Report TM-4513.
MCBRIDE B., M. ZEHE, and S. GORDON (2002), ‘‘NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species,’’ NASA TP-2002-211556.
OLIKARA, C. and G. L. BORMAN (1975), ‘‘A Computer Program for Calculating Properties of Equilibrium Combustion Products with Some Applications to I.C. Engines,’’ SAE paper 750468.
ROSSINI, E D. (1953), Selected Values of Physical and Thermodynamic Properties of Hydrocarbons
and Related Compounds, Carnegie Press, Pittsburgh.
STULL, D. R., E. F. WESTRUM, Jr., and G. C. Singe (1969), The Chemical Thermodynamics of Organic
Compounds, Wiley, New York.
VARGAFTIK, N. B. (1975), Tables on the Thermophysical Properties of Liquids and Gases, Wiley,
New York.
3.9 HOMEWORK
3.1
What is the molecular weight, enthalpy (kJ/kg), and entropy (kJ/(kg K)) of a gas mixture
at 𝑃 = 1000 kPa and 𝑇 = 1000 K, if the mixture contains the following species and mole
fractions?
Species
CO2
H2 O
N2
CO
𝑦𝑖
0.10
0.15
0.70
0.05
3.2
What is the enthalpy (kJ/kg) and entropy (kJ/(kg K)) of a mixture of 30% H2 and 70% CO2
by volume at a temperature of 3000 K and pressure of 2000 K?
3.3
Using the Gordon and McBride equations, Equations 3.38 and 3.39, calculate the enthalpy ℎ̄ and standard entropy 𝑠̄o of CO2 and compare with the gas table values used in
Example 3.1.
3.4
Using the program Fuel.m, at what temperature is the specific heat 𝑐𝑝 of methane CH4 =
3.0 kJ/(kg K)?
3.5
Why does Equation 3.27 contain 𝑦𝑖 ?
3.6
A system whose composition is given below is in equilibrium at 𝑃 = 101 kPa and 𝑇 = 298
K. What is the enthalpy (kJ/kg), specific volume (m3 /kg), and quality 𝜒 of the mixture?
Homework
Species
CO2
H2 O
N2
95
𝑦𝑖
0.125
0.141
0.734
3.7
A four-cylinder four-stroke 2.8 L port injected spark ignition engine is running at 2000 rpm
on a lean (𝜙 = 0.9) mixture of octane and standard air (101 kPa, 298 K). If the octane flow
rate is 2.5 g/s, what is the mass of fuel entering each cylinder per cycle and the volumetric
efficiency?
3.8
An engine cylinder has a 90 mm bore and a 85 mm stroke, and contains air and residual
gases at 350 K and 1 bar. If the engine is to operate on diesel fuel and run lean with an
overall equivalence ratio of 𝜙 = 0.7, what is the mass of diesel fuel that needs to be injected
during the compression stroke? Assume 𝑓 = 0.015, where 𝑓 is the ratio of the residual mass
𝑚r to the cylinder mass 𝑚 prior to fuel injection, and the gas constant 𝑅 of the air--residual
gas mixture = 0.29 kJ/(kg K).
3.9
Using the low-temperature combustion equations, what are the composition, enthalpy, and
entropy of the combustion products of methanol, CH3 OH, at 𝜙 = 1.1, 𝑇 = 1200 K, and
𝑃 = 101 kPa? Compare with the results from the program ecp.m.
3.10
What are the mole fractions of CO2 , H2 O, CO, N2 , and H2 produced when methane (CH4 )
is burned in rich conditions at 𝜙 = 1.1, 𝑇 = 1000 K, and 𝑃 = 101 kPa?
3.11
If a lean (𝜙 = 0.8) mixture of methane CH4 is burned at a temperature of 1500 K and
pressure of 500 kPa, what are the mole fractions of the products, and the product enthalpy,
entropy, and specific heat? Use the program ecp.m.
3.12 (a) At what temperature is the saturation pressure 𝑃sat of octane equal to 0.5 bar? At that
temperature, what is the enthalpy of vaporization ℎ̄ fg ?
(b) Repeat the calculations for tetradecane.
3.13
Compare the enthalpies of vaporization ℎ̄ fg (MJ/kg) of nitromethane, methanol, octane,
and tetradecane at 400 K. What is an advantage of a high enthalpy of vaporization for an
engine fuel?
3.14
A rich (𝜙 = 1.1) mixture of diesel fuel is burned at a temperature of 2000 K and pressure of
750 kPa. Using the program ecp.m, (a) What are the mole fractions of the products, and the
product enthalpy, entropy, specific volume, and specific heat? (b) Repeat the calculation
for 𝜙 = 1.25. Discuss the effect of equivalence ratio.
3.15
Using the program ecp.m, plot the product equilibrium mole fractions as a function of
equivalence ratio (0.5 < 𝜙 < 2) resulting from the combustion of methane at 5000 kPa and
2500 K.
3.16
Derive Equations 3.74 and 3.75 for the species mole fractions of a mixture of air and
residual gas.
3.17
At what equivalence ratio for octane--air mixtures does the carbon to oxygen ratio of the
system equal one? Why is this of interest?
3.18
At what temperature is the concentration of H2 a minimum for the combustion of gasoline
and air at 𝜙 = 1.2 and 4500 kPa? What is the minimum value of H2 ?
96
Fuel, Air, and Combustion Thermodynamics
3.19
At what equivalence ratio is the concentration of OH a maximum for the combustion of
diesel and air at 𝑇 = 2500 K and 4500 kPa? What is the maximum value of OH?
3.20
At what temperature does the mole fraction of NO reach 0.010 for the equilibrium products
resulting from the combustion of gasoline and air at 𝜙 = 1.0 and 5000 kPa?
3.21
At what temperature does the mole fraction of CO reach 0.080 for the equilibrium products
resulting from the combustion of methane and air at 𝜙 = 1.1 and 3000 kPa?
3.22
What is the equilibrium and the frozen specific heat 𝑐𝑝 of the combustion products of
gasoline at a pressure of 2000 kPa and temperature of 2000 K burned at (a) an equivalence
ratio of 1.1, and (b) an equivalence ratio of 0.9?
Chapter
4
Fuel--Air Combustion
Processes
4.1 INTRODUCTION
In this chapter, we apply the first law of thermodynamics to fuel--air combustion processes
in a control volume, and compute the change in state, and the work and heat interactions between the fuel--air mixture and the environment. Using equilibrium combustion modeling,
we are able to determine the product equilibrium state and thermodynamic properties that
result from burning a fuel--air mixture as a function of initial conditions, such as pressure,
temperature, equivalence ratio, and residual fraction. We introduce the heat of combustion, the adiabatic flame temperature, and then examine isentropic processes of a fuel--air
mixture. A second law analysis is performed to introduce maximum work and exergy, and
determine the first and second law efficiencies.
A set of fuel--air cycle computer models, including a closed system Otto cycle, a fourstroke open system Otto cycle, and a homogeneous two-zone finite energy release model
are developed in this chapter. With these models, more realism is introduced into engine
performance modeling, as the models are able to address the effects of parameters such as
equivalence ratio, compression ratio, intake/exhaust pressure ratio, and residual fraction on
net work, imep, and thermal efficiency.
4.2 COMBUSTION AND THE FIRST LAW
With chemical equilibrium modeling, we are able to predict the equilibrium state that
results from burning a fuel--air mixture as a function of initial conditions, such as pressure,
temperature, equivalence ratio, and residual fraction. In this and the next section, we
apply the first law of thermodynamics to fuel--air combustion processes, and compute
energy interactions between the fuel--air mixture and the environment. We discuss constant
pressure and constant volume combustion to illustrate the principles, and introduce the heat
of combustion and the adiabatic flame temperature.
Let us first consider the case in which combustion occurs at constant pressure. Suppose
that the reactants consisting of fuel, air, and residual gases are premixed to a homogeneous
state and burned in a combustion system. Application of the closed system first law to this
combustion process leads to
𝑄 − 𝑊 = 𝑈p − 𝑈r
(4.1)
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
97
98
Fuel--Air Combustion Processes
Since the process is constant pressure, 𝑊 = 𝑃 Δ𝑉 , and since 𝐻 = 𝑈 + 𝑃 𝑉 , we have
(4.2)
𝑄 = 𝐻p − 𝐻r
where the subscript 𝑝 represents products, and the subscript 𝑟 represents reactants. The
enthalpy of the products is equal to the enthalpy of the reactants minus any heat transferred
out of the system.
For an open combustion system, we adopt a control volume approach. The control
volume energy equation indicates that the enthalpy of the products is equal to the enthalpy
of the reactants plus any heat transferred into the system, minus the shaft work out of the
system.
Heat of Combustion
The heat of combustion or heating value, 𝑞c , of a fuel is defined as the heat energy transferred
out of a system per unit mass or mole of fuel when the initial and final states are at the
reference temperature and pressure, 𝑇o = 298.15 K and 𝑃o = 101 kPa. The number of moles
in the system is not constant during a combustion change of state. By convention, the heat
transferred out of a system is negative, and the heat of combustion is a positive number,
that is, 𝑞c = −𝑄, so
(4.3)
𝑞c = 𝐻r − 𝐻p
An analogous discussion could be presented for constant volume combustion where 𝑊 =
𝑃 Δ𝑉 = 0 and
(4.4)
𝑞c = 𝑈r − 𝑈p
However, as a rule of thumb, when the heat of combustion referred to without qualification,
constant pressure combustion is implied.
Furthermore, the combustion is assumed to be complete, with the fuel burning to
carbon dioxide and water. Since the products are at low temperature (𝑇o < 1000 K) , the
analyses that led to Table 3.6 for the lean or stoichiometric case can be used to compute
the heat of combustion 𝑞c . In this case, it can be shown that Equation 4.3 becomes
∑
∑
𝑞c =
𝑛p,i ℎ̄ p,i
𝑛r,i ℎ̄ r,i −
i
i
= 1 × ℎ̄ fuel − 𝑛CO2 ℎ̄ CO2 − 𝑛H2 O [ℎ̄ H2 O − (1 − 𝜒)ℎ̄ fg,H2 O ]
(kJ∕kmolfuel )
(4.5)
since the enthalpies of oxygen O2 and nitrogen N2 are assigned to be zero at the reference
temperature 𝑇o = 298.15 K.
Two values of 𝑞c are recognized (1) the lower heat of combustion 𝑞lhc is defined as the
state where all of the water in the products is vapor (the quality 𝜒 = 1 ), and (2) the higher
heat of combustion, 𝑞hhc is defined as the state where all of the water in the products is
liquid (the quality 𝜒 = 0 ). If the water quality is not specified, one usually assumes that
𝑞c = 𝑞lhc .
The lower 𝑞lhc and higher heat of combustion 𝑞hhc of several gaseous and liquid fuels
are given in Table 4.1. The heat of combustion is used primarily in two ways (1) in some
cases, such as in the gas cycles of Chapter 2 or when solving reacting Navier--Stokes
equations, it is desirable to relax the rigor of the thermodynamics by using the heat
of combustion to define an equivalent energy release; and (2) for practical fuels, as
discussed in Chapter 10, the enthalpy at 𝑇o = 298.15 K can be determined inexpensively
by measurement of the heat of combustion. Table 4.1 also lists ℎ̄ of , the enthalpy of
Combustion and the First Law
99
Table 4.1 Enthalpy of Formation, Entropy, Lower/Higher Heat of Combustion, and Maximum
Available Energy of Combustion1
Fuel
CH4 (g)
C3 H8 (g)
C7 H17 (l)
C8 H18 (l)
C14.4 H24.9 (l)
C15 H32 (l)
CH4 O (l)
C2 H6 O (l)
CH3 NO2 (l)
H2 (g)
C2 H2 (g)
C2 N2 (g)
NH3 (g)
C6 H6 (l)
C10 H8 (s)
C (s)
C176 H144 O8 N3 (s)
Methane
Propane
Gasoline2
Octane
Diesel2
Pentadecane
Methanol
Ethanol
Nitromethane
Hydrogen
Acetylene
Cyanogen
Ammonia
Benzene
Naphthalene
Graphite
Coal2
ℎ̄ of
(MJ/kmol)
𝑠̄o298
(kJ/(kmol K))
𝑞lhc
(MJ/kg)
𝑞hhc
(MJ/kg)
𝑎o
(MJ/kg)
−74.9
−103.9
−305.6
−249.96
−174.0
−428.9
−239.1
−277.2
−113.1
0.0
226.7
309.1
−45.7
48.91
78.1
0.0
−10,000.0
186.2
269.9
345.8
360.8
525.9
587.5
126.8
160.7
171.8
130.6
200.8
241.5
192.6
173.0
166.9
5.7
3000.0
50.01
46.36
44.51
44.43
42.94
43.99
19.91
26.82
10.54
119.95
48.22
21.06
18.61
40.14
38.86
32.76
31.57
55.5
50.3
52.42
49.16
47.87
47.67
45.73
47.22
22.68
29.71
12.43
119.52
48.58
21.29
20.29
42.14
40.84
33.70
33.57
47.9
47.3
22.7
29.7
11.6
141.6
49.9
21.0
22.5
40.3
1 Based
on equilibrium water quality, lean combustion at 𝜙 = 0.01, 𝑇o = 298 K, 𝑃o = 1.013 bar and unmixed
reactants.
2 Estimated for typical fuel.
formation, 𝑠̄o298 , the absolute entropy at 298 K, and 𝑎o , the maximum available energy of
combustion.
EXAMPLE 4.1
Heat of Combustion
Compare the lower, 𝑞lhc , and higher, 𝑞hhc , heat of combustion of cetane C16 H34 to the heat
released 𝑞eq with equilibrium water quality products.
SOLUTION Assume standard reference conditions 𝑃 = 𝑃o = 1 atm and 𝑇o = 298.15 K for reactants and products, and the air and fuel enter unmixed. The molecular mass 𝑀 of cetane
is 226.4. As shown in Appendix E.6, the equilibrium water quality 𝜒eq for this reaction is
0.208. The nitrogen enthalpy is zero for both the reactants and products, and is not included
in the energy equation computation.
The stoichiometric combustion equation per mole of cetane is
1C16 H34 + 24.5 O2 → 16 CO2 + 17 H2 O
The first law, Equation 4.3, per kmol of fuel, is
∑
∑
𝑞̄c =
𝑛r,i ℎ̄ r,i −
𝑛p,i ℎ̄ p,i
i
i
= 1 × ℎ̄ fuel − 𝑛CO2 ℎ̄ CO2 − 𝑛H2 O [ℎ̄ H2 O − (1 − 𝜒)ℎ̄ fg,H2 O ]
= + 1 (−373.34) − 16 (−393.52) − 17 [(−241.83 − (1 − 𝜒)(43.99))]
100
Fuel--Air Combustion Processes
= + 10, 034 + (747.8)(1 − 𝜒) (MJ∕kmol)
Therefore,
𝑞̄c (𝜒 = 1)
= 10, 034∕226.4 = 44.4 MJ∕kg
𝑀
𝑞̄ (𝜒 = 0.208)
= c
= 10, 624∕226.4 = 47.0 MJ∕kg
𝑀
𝑞̄ (𝜒 = 0)
= c
= 10, 782∕226.4 = 47.7 MJ∕kg
𝑀
𝑞lhc =
𝑞eq
𝑞hhc
Adiabatic Flame Temperature
Another useful combustion parameter is the adiabatic flame temperature 𝑇f . It is defined
as the temperature of the combustion products when completely burned with no shaft work
(𝑊 = 0) or heat transfer (𝑄 = 0) to the surroundings. The adiabatic flame temperature
represents the maximum temperature of a combustion process, since any heat transfer from
the reaction and incomplete combustion will lower the temperature of the products.
For constant pressure combustion, the first law is
𝐻p = 𝐻r
(4.6)
The initial state of the reactants is assumed to be at the reference temperature and pressure,
𝑇o = 298.15 K and 𝑃o = 101 kPa. Since the product temperature is generally unknown in
first law combustion calculations, iteration with an initial temperature estimate is required
to determine the product mixture enthalpy. An assumption also needs to be made about
the amount of dissociation in the combustion products. The adiabatic flame temperature
with dissociation of the combustion products will be less than the computed value for
combustion without dissociation.
For constant volume combustion, the first law of thermodynamics with no work
(𝑊 = 0) or heat transfer (𝑄 = 0) to the surroundings is
𝑈p = 𝑈r
(4.7)
The constant volume adiabatic flame temperature is greater than the constant pressure
adiabatic flame temperature since the 𝑝𝑑𝑉 work is zero in a constant volume process.
The stoichiometric adiabatic flame temperature of several fuels is listed in Table 4.2. The
results tabulated assume reference conditions of 𝑇o = 298.15 K, 𝑃o = 101 kPa, 𝑓 = 0,
and 𝜙 = 1.0. There is little dependence on fuel type among the hydrocarbons, which have
adiabatic flame temperatures of about 2250 K. Note that operating engines will have a
larger 𝑇o , and a corresponding larger adiabatic flame temperature.
EXAMPLE 4.2
Adiabatic Flame Temperature
A stoichiometric mixture of gasoline C7 H17 , air, and residual gas is burned at constant
pressure. Given that 𝑇1 = 298 K, 𝑃1 = 101.3 kPa, and 𝑓 = 0.10, what is the constant
pressure adiabatic flame temperature?
SOLUTION Appendix F.12 contains a listing of the program AdiabaticFlameTemp.m that
computes the constant pressure adiabatic flame temperature. The program uses the fuel-air--residual gas routine farg.m, the equilibrium combustion routine ecp.m detailed in
the previous chapter, and Newton--Raphson iteration for constant pressure, with inputs the
Combustion and the First Law
101
Table 4.2 Stoichiometric Adiabatic Flame Temperature of Various Fuels
Formula
CH4 (g)
C3 H8 (g)
C8 H18 (l)
C15 H32 (l)
C20 H40 (g)
CH4 O (l)
C2 H6 O (l)
CH3 NO2 (l)
H2 (g)
C2 N2 (g)
NH3 (g)
C2 H2 (g)
C10 H8 (s)
Fuel
𝑇f (K)
Methane
Propane
Octane
Pentadecane
Eicosane
Methanol
Ethanol
Nitromethane
Hydrogen
Cyanogen
Ammonia
Acetylene
Naphthalene
2227
2268
2266
2269
2291
2151
2197
2545
2383
2596
2076
2540
2328
𝑃o =1.0 atm, 𝑇o =298.15 K, 𝑓 = 0.0
pressure 𝑃 , initial temperature 𝑇 , fuel--air equivalence ratio 𝜙, and residual mass fraction
𝑓 . The available fuel types are methane (CH4 ), gasoline (C7 H17 ), diesel (C14.4 H24.9 ),
methanol (CH3 OH), and nitromethane (CH3 NO2 ).
The above information is entered into the program as shown below:
% Computes const pressure adiabatic flame temperature
% Inputs:
T1 = 298;% initial temperature (K)
P1 = 101.3; % initial pressure (kPa)
PHI = 1.1; %
f = 0.1; %
equivalence ratio
residual fraction
ifuel=2; % 1=Methane, 2=Gasoline, 3=Diesel, 4=Methanol, ...
5=Nitromethane
The resulting adiabatic flame temperature is 𝑇 = 2094 K.
The effect of equivalence ratio is plotted in Figure 4.1, which indicates that the adiabatic
flame temperature is maximum near stoichiometric. This is consistent with the effect of
equivalence ratio on enthalpy, as shown in the Chapter 3, since mixtures with a relatively
lower specific heat will undergo a larger temperature change for a given energy release.
Additional calculations show the adiabatic flame temperature decreases with increasing
residual fraction, (see homework problem at end of chapter), and slightly increases with
pressure. Note that of the fuels plotted, ethanol consistently has the lowest and nitromethane
has the greatest adiabatic flame temperature.
Isentropic Processes
In internal combustion engine modeling, we need to determine the change in state due to an
assumed isentropic compression or expansion to a specified pressure or specific volume.
With a known change from an initial state 1 to a final state 2, the first law for an closed
102
Fuel--Air Combustion Processes
Figure 4.1 Adiabatic
flame temperature of
some fuels initially at
atmospheric pressure
and temperature,
(𝑓 = 0.0).
Adiabatic flame temperature—Tf (K)
3000
CH3NO2
2500
H2
C8H18
2000
C2H6O
1500
1000
P1 = 1.0 atm
T1 = 298 K
500
0.2
0.4
0.6
0.8
1.0
Equivalence ratio
1.2
1.4
system can be used to determine the work transfer 𝑊1−2 :
−𝑊1−2 = 𝑈2 − 𝑈1
(4.8)
and for an open system, the first law is
−𝑊1−2 = 𝐻2 − 𝐻1
(4.9)
For a mixture of ideal gases that chemically reacts to changing constraints, such as the
volume and pressure relationship in an isentropic process, 𝑝𝑉 𝛾 = constant, simple algebraic
relationships between the initial and final state cannot be derived, and computer solution
is required. The equilibrium constant methodologies discussed in the previous sections
allow the determination of the properties of mixture of gases given their temperature and
pressure, so for an isentropic change of volume where the final temperature is unknown,
iteration is required.
Given the heat transferred from a control volume, and the combustion pressure 𝑃 ,
we can use Equation 4.2 to solve for the product enthalpy 𝐻p . With two thermodynamic
variables, 𝑃 and 𝐻, known, the other properties such as the temperature 𝑇 , specific
volume 𝑣, and internal energy 𝑈 of the products can also be computed. The equilibrium
constant method of Section 3.6 is formulated with the assumption that the product pressure
and temperature are known. Since the product temperature is generally unknown in first
law combustion calculations, iteration with an initial temperature estimate is required.
EXAMPLE 4.3
Isentropic Fuel--Air Processes
A gasoline fuel--air mixture with 𝜙 = 0.8 is initially at 𝑇1 = 300 K, 𝑃1 = 101.3 kPa. The
mixture is compressed isentropically to state 2 to pressure 𝑃2 = 2020 kPa. (a) What is the
temperature 𝑇2 and the work 𝑤1−2 ? (b) What is the compression ratio?
SOLUTION Since the engine cylinder volume is a closed system, the first law on a per unit mass
of mixture basis for this isentropic process is
−𝑤1−2 = 𝑢2 − 𝑢1
(4.10)
Maximum Work and the Second Law
103
Using the Equilibrium Combustion Solver program RunEcp.m, the mixture properties
at state 1 at the beginning of compression are
𝑇1 = 300 K
𝑃1 = 101.25 kPa
ℎ1 = −2364 kJ∕kg
𝑢1 = −2451 kJ∕kg
𝑠1 = 7.040 kJ∕kg K
𝑣1 = 0.862 m3 ∕kg
Since the compression is isentropic, 𝑠2 = 𝑠1 =7.040 kJ/(kg K). The pressure at state 2
is known, but the temperature is not, so iteration of the temperature input to the program
is needed, keeping entropy constant at 𝑠2 = 𝑠1 =7.040 kJ/(kg K). This procedure results in
𝑇2 = 660.5 K, 𝑢2 = −2156 kJ/kg, and 𝑣2 = 0.095 m3 /kg. Therefore, the compression work
−𝑤1−2 is equal to 295 kJ/kg, and the compression ratio 𝑟 is 9.07.
4.3 MAXIMUM WORK AND THE SECOND LAW
For the purpose of defining engine efficiency with actual fuels, we will need to determine
the maximum work that can be done by an engine system as it changes state. The following
analysis derives both closed and open system maximum work in terms of the change in
exergy.
For a closed system such as the compression stroke of a piston--cylinder system in
communication with the environment at 𝑇o , 𝑃o , the first and second laws of thermodynamics
for a change in state from 1 to 2 are
𝑄1−2 − 𝑊1−2 = 𝑈2 − 𝑈1
(4.11)
𝑄1−2 ≤ 𝑇o (𝑆2 − 𝑆1 )
(4.12)
The total work transfer 𝑊1−2 can be therefore be expressed as
[
]
𝑊1−2 ≤ − (𝑈2 − 𝑈1 ) − 𝑇o (𝑆2 − 𝑆1 )
(4.13)
if we subtract the work done by the system against the atmosphere, 𝑃o (𝑉2 − 𝑉1 ), we obtain
the actual work
[
]
(4.14)
𝑊act,1−2 ≤ − (𝑈2 − 𝑈1 ) + 𝑃o (𝑉2 − 𝑉1 ) − 𝑇o (𝑆2 − 𝑆1 )
We define the nonflow exergy 𝐴, a property of the system for a fixed 𝑇o , 𝑃o , as
𝐴 = 𝑈 + 𝑃o 𝑉 − 𝑇o 𝑆
(4.15)
𝑎 = 𝑢 + 𝑃o 𝑣 − 𝑇o 𝑠
(4.16)
𝑊act,1−2 ≤ −(𝐴2 − 𝐴1 )
(4.17)
and per unit mass,
so
Therefore, the maximum work, 𝑊max,1−2 , that can be done by the system as it changes state
from 1 to 2 is the change in the exergy 𝐴:
[
]
(4.18)
𝑊max,1−2 = −(𝐴2 − 𝐴1 ) = − (𝑈2 − 𝑈1 ) + 𝑃o (𝑉2 − 𝑉1 ) − 𝑇o (𝑆2 − 𝑆1 )
104
Fuel--Air Combustion Processes
and per unit mass,
[
]
𝑤max,1−2 = −(𝑎2 − 𝑎1 ) = − (𝑢2 − 𝑢1 ) + 𝑃o (𝑣2 − 𝑣1 ) − 𝑇o (𝑠2 − 𝑠1 )
(4.19)
For a compression process, the change in exergy will be positive, since work is performed
on the system, for a exothermic combustion process the change in exergy will be negative,
due to the change in chemical composition of the system, and for an expansion process, the
change in exergy will be negative, since work is performed by the system.
If the system comes to thermal and mechanical equilibrium with the atmosphere during
a change in state from 1 to 0, where the subscript 0 represents the final state in equilibrium
with the atmosphere, then the maximum work is
𝑊max,1−0 = 𝐴1 − 𝐴0
(4.20)
Exergy Change for an Isentropic Compression or Expansion
If a change in state for a closed system is isentropic, then Equation 4.19 reduces to
𝑎2 − 𝑎1 = (𝑢2 − 𝑢1 ) + 𝑃o (𝑣2 − 𝑣1 )
(4.21)
For an ideal gas with constant properties, the change in state for an isentropic compression
or expansion is
𝑃 𝑣𝛾 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
(4.22)
Using the definition of 𝑐v , we can relate 𝑢2 − 𝑢1 to 𝑇2 − 𝑇1
𝑢2 − 𝑢1 = 𝑐v (𝑇2 − 𝑇1 )
(4.23)
and if 𝑟 = 𝑣1 ∕𝑣2 , and 𝑣 = RT/P, then upon substitution into Equation 4.21, the exergy
change for an ideal gas undergoing an isentropic compression from state 1 to state 2 is
]
[
𝑃o
1
𝛾−1
− 1) + (𝛾 − 1)( − 1)
(4.24)
𝑎2 − 𝑎1 = 𝑐v 𝑇1 (𝑟
𝑃1
𝑟
EXAMPLE 4.4
Isentropic Compression of a Fuel--Air Mixture
What is the exergy change for the isentropic fuel--air compression of Example 4.3?
SOLUTION Since the process is isentropic, 𝑠2 = 𝑠1 , so
𝑎2 − 𝑎1 = (𝑢2 − 𝑢1 ) + 𝑃o (𝑣2 − 𝑣1 )
= −2156 − (−2451) + (101.25)(0.095 − 0.862)
(4.25)
= 217.3 kJ∕kg.
Comment: The change in exergy is less than the isentropic compression work, since we
have subtracted the work done against the atmosphere.
Available Energy of Combustion
The appropriate definition of efficiency for any of the gas cycles presented in Chapter 2 is
clear, since the efficiency for a gas cycle is defined as the fraction of an ‘‘equivalent heat
transfer’’ which is converted to work. When the analysis takes into account that the fuel
is burned rather than heat being transferred to produce work, the first law efficiency for
Maximum Work and the Second Law
105
⋅
Qc.v.
Air
Po, To
Cyclic
engine
Exhaust
Po, To
Fuel
Figure 4.2 A control Po, To
volume for analyzing
the maximum work of
a cyclic engine.
Control surface
⋅
W
c.v.
a control volume (c.v.) is defined as the ratio of the net work done per unit mass of fuel
inducted into the cylinder, 𝑤c.v. , to the heat of combustion, 𝑞c .
(4.26)
𝜂 = 𝑤c.v. ∕𝑞c
Internal combustion engine efficiency can also be defined from the perspective of
the second law of thermodynamics. While the first law takes the energy transfers to the
surroundings by way of the coolant and the exhaust into account, it does not consider the
maximum possible work. The second law definition of engine efficiency, 𝜂II , is the ratio of
the net work done by the engine to the maximum possible work:
(4.27)
𝜂II = 𝑊c.v. ∕𝑊max
Following Obert (1973), the maximum possible work, 𝑊max is found from application of
the first and second law to the control volume shown in Figure 4.2. Note that the fuel and air
reactants (r) flow into the engine at 𝑃o , 𝑇o , and the combustion products (𝑝) are exhausted
from the engine at 𝑃o , 𝑇o .
)
(
∑
∑
𝑑𝐸
𝑄̇ c.v. − 𝑊̇ c.v. =
+
𝑚̇ ℎ −
𝑚̇ ℎ
(4.28)
𝑑𝑡 c.v.
p
r
Let us integrate over one period of the engine’s cycle
∑
∑
𝑄c.v. − 𝑊c.v. =
𝑚ℎ −
𝑚ℎ
p
(4.29)
r
The maximum work is obtained only if the process is reversible, in which case the second
law applied to the control volume is an equality:
)
(
∑
∑
𝑚𝑠
(4.30)
𝑚𝑠 −
𝑄c.v = 𝑇o
p
r
The only way in which the reversible heat transfer of Equation 4.30 can occur between
an engine and its surroundings is via an intervening Carnot engine. Upon substitution of
Equation 4.30 into Equation 4.28, the maximum work is
)
(
∑
∑
∑
∑
𝑚ℎ −
𝑚ℎ + 𝑇o
𝑚𝑠 −
𝑚𝑠
𝑊max =
r
p
p
= (𝐻r − 𝐻p ) − 𝑇o (𝑆r − 𝑆p )
= 𝐵r − 𝐵p
r
(4.31)
106
Fuel--Air Combustion Processes
where the parameter 𝐵 is the flow exergy
𝐵 = 𝐻 − 𝑇o 𝑆
(4.32)
Since the reactants and the products are both at reference conditions, the maximum
work can also be expressed as the change in the Gibbs free energy, 𝐺:
𝑊max = 𝐺r − 𝐺p
(4.33)
The available energy of combustion 𝑎c is defined as the maximum work per unit mass (or
mole) of fuel
𝑎c = 𝑊max ∕𝑚f
=
1
[(𝐻r − 𝐻p ) + 𝑇o (𝑆p − 𝑆r )]
𝑚f
(4.34)
= (𝐵r − 𝐵p )∕𝑚f
so the second law efficiency can be expressed as
𝜂II =
𝑊c.v.
𝑚f 𝑎 c
(4.35)
The difference between the available energy of combustion, Δ(𝐻 − 𝑇o 𝑆), and the heat of
combustion, Δ𝐻, is that the available energy of combustion takes into account the change
in entropy due to changes in composition of reactants.
EXAMPLE 4.5
Heat of Combustion and Available Energy of Combustion
Compare the available energy of combustion, 𝑎c with the lower heat of combustion, 𝑞lhc ,
for the stoichiometric combustion of methane at standard reference conditions.
SOLUTION Assume standard reference conditions 𝑃 = 𝑃o = 1 bar and 𝑇o = 298.15 K, the air
and fuel enter unmixed, and that the products leave in a gaseous state with a quality 𝜒 = 1.
Since the nitrogen enthalpy is zero for both the reactants and products it is not included in
the computation. The chemical combustion equation per mole of methane is
1 CH4 + 2 O2 → CO2 + 2 H2 O
and the molecular mass 𝑀 of methane is 16.04.
Equations 4.3 and 4.34, per unit mass of fuel, are
𝑞c = 𝐻r − 𝐻p
𝑎c = 𝐵r − 𝐵p = (𝐻r − 𝐻p ) − 𝑇o (𝑆r − 𝑆p )
From the tabular data for ℎ̄of and 𝑠̄oi given in Table 4.1,
∑
𝐻r =
𝑛i ℎ̄i
ri
= (1)(−74.87) + 0.21(0) = −74.87 MJ
∑
𝑛i ℎ̄i
𝐻p =
pi
Maximum Work and the Second Law
107
= (1)(−393.5) + (2)(−241.8) = −877.17 MJ
∑
𝑆r =
𝑛i [𝑠̄oi − 𝑅u ln𝑦i )]
ri
= (1)[186.2 − 8.314 ln(1∕1)] + (2)[205.15 − 8.314 ln(0.21∕1)] = 0.622 MJ/K
∑
𝑛i [𝑠̄oi − 𝑅u ln𝑦i )]
𝑆p =
pi
= (1)[213.79 − 8.314 ln(0.0004∕1)] + (2)[188.83 − 8.314 ln(0.02∕1)] = 0.721 MJ/K
Therefore,
𝑞c = 𝐻r − 𝐻p
= −74.87 − (−877.17)
= 802.3 MJ/kmol = 50.01 MJ/kg
𝑎c = (𝐻r − 𝐻p ) − 𝑇o (𝑆r − 𝑆p )
= 802.3 − (298)(0.622 − 0.721)
= 831.8 MJ/kmol = 51.86 MJ/kg
Comment: At 298 K, N2 and O2 are assigned zero enthalpy, so the inclusion of N2 will not
change the results.
Figure 4.3 plots the available energy of combustion and the heat of combustion as a
function of equivalence ratio at 𝑇o = 298 K, 𝑃o = 1 atm. The maximum work is attained
only if the exhaust is in equilibrium at the state 𝑇o , 𝑃o . This also implies that each exhaust
species such as CO2 and H2 O is at the partial pressure that it is found in the environment.
The exhaust water quality is evaluated by setting the partial pressure of the vapor equal to
the saturation vapor pressure at 𝑇 = 𝑇o .
It is evident from Figure 4.3 that more energy is available per unit mass of fuel if an
engine is fueled lean than if it is fueled rich. In the rich case, there is significant carbon
monoxide and hydrogen in the exhaust. Thus, not all of the fuel’s chemical energy is
released and the exhaust gases could, in principle, be used as a fuel for some other engine.
In practice, however, those gases are usually exhausted to the atmosphere and the energy is
wasted. For this reason, in this chapter, we will base our fuel--air cycle thermal efficiency on
the maximum available energy of combustion that occurs for very lean equivalence ratios.
The value of 𝜙 = 0.01 was chosen as being close enough to zero for practical purposes.
Thus, letting
𝑎o = 𝑎c,𝜙=0.01
(4.36)
We define the second law thermal efficiency for a fuel--air cycle to be
𝜂II =
𝑊c.v.
𝑤 (1 + 𝜙𝐹 𝐴s )
= c.v.
𝑚f 𝑎 o
(1 − 𝑓 )𝜙𝐹 𝐴s 𝑎o
(4.37)
where 𝑤c.v. is the work per unit mass done by the system.
To a certain extent, the definition of thermal efficiency is equivocal. It seems impractical to take into account the small amount of work that can in principle be realized because
the exhaust composition is different than that of the atmosphere. One could make a case for
defining efficiency in terms of the heat of combustion on the premise that never will a heat
108
Fuel--Air Combustion Processes
50
Energy (MJ/kg)
40
Figure 4.3 Available energy
and heat of combustion for
liquid gasoline and methanol.
Fuel and air are unmixed.
Products are mixed with
equilibrium water quality.
𝑃o = 1.013 bar, 𝑇o = 298 K.
30
C7H17
(gasoline)
20
CH3OH
(methanol)
Heat of combustion
10
Available energy of combustion
0
0.2
0.4 0.6 0.8 1.0
Equivalence ratio
1.2
1.4
1.6
engine of any sort, including the Carnot engine, be used to reduce the irreversibilities associated with the heat transfer. By inspection of Figure 4.3, one would use the stoichiometric
heat of combustion, for here the heat of combustion is maximum. Arguments can also
be made for use of either the lower or the higher heat of combustion. Additionally, from
Table 4.1, notice that for the most part there is little difference between the lower heat of
combustion 𝑞lhc and the maximum available energy 𝑎o . Therefore, many engineers prefer
to measure engine efficiency using the specific fuel consumption because its definition is
unequivocal.
4.4 FUEL--AIR OTTO CYCLE
We now combine the thermodynamic processes discussed in previous sections with the
fuel--air equations of state to form a fuel--air cycle analysis to compute the thermal efficiency, work, and imep produced by an internal combustion engine. A fuel--air cycle
model includes the effect of the change in composition of the fuel--air mixture as a result
of combustion. During compression, the gases in the cylinder are a mixture of air, fuel,
and residual exhaust gas, and during expansion, the gases in the cylinder are equilibrium
combustion products. Using these fuel--air combustion models, it is possible to compute the
properties at states corresponding to the beginning and end of compression, combustion,
and expansion for given fuel--air mixtures.
We start with a simple fuel--air Otto cycle in which the combustion process is assumed to be constant volume at top dead center. In a subsequent section, we develop a
fuel--air finite energy release model in which the combustion occurs over a given change
Fuel--Air Otto Cycle
109
in crank angle. The groundwork for introducing fuel--air cycles was laid in Chapter 2,
where fundamental thermodynamic processes were presented; and in Chapter 3, where
the thermodynamic properties and equations of state for equilibrium fuel--air--exhaust gas
mixtures were developed.
Since the combustion process is assumed to be adiabatic and constant volume in an
Otto cycle, the internal energy is constant, so 𝑢3 = 𝑢2 , and the increase in temperature 𝑇
and pressure 𝑃 is due to the change in chemical composition from an unburned fuel air
mixture to an equilibrium combustion product mixture. The basic processes of a fuel--air
Otto cycle necessary to compute the efficiency and the indicated mean effective pressure
are
isentropic compression from 𝑣1 to 𝑣2 , with 𝑠2 = 𝑠1
Adiabatic, constant volume combustion, 𝑣 = constant, with 𝑢3 = 𝑢2
Isentropic expansion from 𝑣3 to 𝑣4 , with 𝑠4 = 𝑠3
1 to 2
2 to 3
3 to 4
The work of the fuel--air Otto cycle is
𝑤net = (𝑢3 − 𝑢4 ) − (𝑢2 − 𝑢1 ) = 𝑢1 − 𝑢4
(4.38)
and the imep is
imep =
𝑤net
𝑣1 − 𝑣2
(4.39)
The above equations are solved numerically in the fuel--air Otto cycle program
OttoFuel.m listed in Appendix F.13. The engine parameters that are input to the Otto
fuel--air cycle program are the compression ratio 𝑟, the fuel--air equivalence ratio 𝜙, the
residual mass fraction 𝑓 , the fuel type, and the initial mixture temperature 𝑇1 and pressure
𝑃1 . Using the FARG and ECP routines to determine residual fraction and equilibrium
properties, the program computes the mixture temperature, pressure, enthalpy, specific
volume, and specific heat at each of the four states.
EXAMPLE 4.6
Fuel--Air Otto Cycle
Compute the state properties, work, imep, and thermal efficiency of a fuel--air Otto cycle
with the following initial conditions: gasoline fuel with 𝑃1 = 101.3 kPa, 𝑇1 = 350 K, 𝜙 =
1.1, residual fraction 𝑓 = 0.1, and a compression ratio 𝑟 = 10.
SOLUTION As described in detail in Appendix F.13, the fuel--air Otto cycle program OttoFuel.m computes the mixture properties at the four states, as well as the work, imep, and
thermal efficiency.
The program input is
% program OttoFuel - computes const vol fuel air cycle
%
first,isentropic compression from v1 to known v2
% establish initial conditions at state 1
clear;
T1 = 350;
%Kelvin
P1 = 101.3;
%kPa
phi = 1.1;
%equivalence ratio
f= 0.1;
%residual fraction
Fuel--Air Combustion Processes
rc=10.;
%compression ratio
...
The program output is
Ottofuel input conditions: phi=
State
-------
1
1.10
----
2
fuel=
----
2
3 ----
4
Pressure (kPa) =
101.3
2113.1
8708.0
Temperature (K) =
350.0
730.1
2779.7
1604.9
Enthalpy(kJ/kgK) = -390.8
47.3
677.8
-1167.3
500.2
Volume (m3/kg) =
0.956
0.096
0.096
0.956
Cp (kJ/kg K) =
1.078
1.23
2.052
1.424
Work (kJ/kg) =
Efficiency =
Imep (kPa) =
1157.8
0.430
1345.7
The maximum temperature and pressure are 𝑇3 = 2780 K and 𝑃3 = 8708 kPa. The
work produced is 1158 kJ/kg, the imep is 1346 kPa, and the thermal efficiency is 0.43.
Additional results obtained for the Fuel--Air Otto cycle model as a function of equivalence ratio, compression ratio, and residual fraction are plotted in Figures 4.4--4.6. The
following are some of the important conclusions:
Indicated thermal efficiency
0.6
Imep/P1
110
Figure 4.4 Effect of equivalence
ratio on Otto fuel--air cycle.
0.5
0.4
r
15
10
0.3
5
0.2
0.1
C7H17, gasoline
P1 = 1.0 bar
T1 = 350 K
0
20
18
16
14
12
10
8
6
4
2
0
r
15
10
5
0.7
0.8
0.9
1.0
1.1
1.2
Fuel–air equivalence ratio
1.3
1.4
1.5
Fuel--Air Otto Cycle
111
(Imep/P1)
Indicated thermal efficiency
0.6
0.75
0.5
1.00
0.4
1.30
0.3
C7H17, gasoline
f = 0.10
P1 = 1.0 bar
T1 = 350 K
0.2
0
0
20
18
16
14
12
10
8
6
4
2
0
Figure 4.5 Effect of compression
ratio on Otto fuel--air cycle.
1.00
1.30
0.75
5
10
15
20
Compression ratio (r)
0.6
(Imep /P1)
Indicated thermal efficiency
r
Figure 4.6 Effect of residual
fraction on Otto fuel--air cycle.
0.5
15
10
0.4
0.3
5
0.2
0
0
20
18
16
14
12
10
8
6
4
2
0
C7H17, gasoline
f = 0.10
P1 = 1.0 bar
T1 = (330 + 200 f ) K
r
12
10
5
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
Residual mass fraction ( f )
112
Fuel--Air Combustion Processes
1. The indicated efficiency increases with increasing compression ratio, is maximized by
lean combustion, and is practically independent of the initial temperature and initial
pressure. In actual engines, maximum efficiency occurs at stoichiometric or slightly
lean; excessive dilution of the charge with air degrades the combustion.
2. The indicated mean effective pressure increases with increasing compression ratio, is
maximized slightly rich of stoichiometric, and increases linearly with the initial density
(i.e., imep ∼ 𝑃1 and imep ∼ 1∕𝑇1 ). The maximum imep at slightly rich equivalence ratio
is due to the dissociation of the exhaust products.
3. For a given compression ratio, the peak pressure is proportional to the indicated mean
effective pressure.
4. Peak temperatures in the cycles are largest for equivalence ratios slightly rich of stoichiometric.
The results shown are characteristic of most hydrocarbon fuels. It is of interest to
explore the influence of fuel properties for some alternative fuels as we look to the future.
Table 4.3 presents results obtained for two different compression ratios and five different
fuels. Notice that there is very little difference among hydrocarbons. According to this
analysis, diesel fuel would be just as good as gasoline in a homogeneous charge spark
ignition engine; in reality, of course, knock would be a problem. Note that nitromethane is
an excellent choice for a racing fuel, as it has the largest imep of the fuels in Table 4.3.
It is also of interest to examine the influence of the residual fraction, since a widely used
technique for emission control is exhaust gas recirculation (EGR). By pumping exhaust
gas into the intake manifold and mixing it with the fuel and air, one has, in essence,
increased the residual gas fraction. Although the exhaust gas so recirculated is cooled
before introduction into the induction system, it is still considerably warmer than the inlet
air. Therefore, we will increase the inlet temperature in our computations simultaneously
to examine the overall effect.
To illustrate, assume the initial temperature at the start of compression is 𝑇1 = 1330 +
200𝑓 (K). The results obtained for gasoline are given in Figure 4.6. Notice that the efficiency
increases slightly with increasing dilution of the charge by residual gas. Notice too that
imep falls with increasing residual fraction 𝑓 ; it falls because the residual gas displaces the
fuel--air mixture volume, and it also warms the fuel--air mixture, thereby reducing the
Table 4.3 Effect of Fuel Type on Otto Fuel--Air Cycle
Fuel
Formula
𝑟
𝜂Otto
imep (bar)
Gasoline
C7 H17
Diesel
C14.4 H24.9
Methane
CH4
Methanol
CH3 OH
Nitromethane
CH3 NO2
10
15
10
15
10
15
10
15
10
15
0.44
0.49
0.44
0.49
0.44
0.49
0.43
0.48
0.39
0.43
13.3
14.4
13.7
14.9
12.2
13.1
13.1
14.2
21.0
23.1
𝜙 = 1.0, 𝑓 = 0.10, 𝑃1 = 1.0 bar, 𝑇1 = 350 K
Four-Stroke Fuel--Air Otto Cycle
113
charge density. As will be discussed later in this chapter, all of the conclusions drawn from
Figures 4.4 to 4.6 apply to actual engines, provided that they are operated at optimum
spark timing.
4.5 FOUR-STROKE FUEL--AIR OTTO CYCLE
In this section, we develop a four-stroke Otto fuel--air cycle with idealized inlet and
exhaust processes. In this case, the input engine parameters 𝑇1 , 𝑃1 , and 𝑓 are no longer
the independent variables. Instead, the intake pressure 𝑃i , the exhaust pressure 𝑃e , and the
intake temperature 𝑇i are the independent variables and used as the input engine parameters.
The additional processes, introduced in Chapter 2, are reiterated here
4 to 5
5 to 6
6 to 7
7 to 1
Constant cylinder volume blowdown
Constant pressure exhaust
Constant cylinder volume reversion
Constant pressure induction
The exhaust blowdown is considered to be isentropic as far as the control mass is
concerned. One solves for the temperature 𝑇5 by requiring that 𝑆5 = 𝑆4 and 𝑃5 = 𝑃e .
Application of the first law to the control mass during exhaust leads to the conclusion that
ℎ6 = ℎ 5
𝑃6 = 𝑃5
𝑇6 = 𝑇5
𝑉6 = 𝑉5
These are still valid conclusions even though now we are treating the exhaust gas as
equilibrium combustion products.
The residual fraction, given by Equation 2.67, is
1 𝑣4
(4.40)
𝑓=
𝑟 𝑣6
The energy equation applied to the cylinder control volume during intake is given by
Equation 2.71. Note that Equation 2.79 is no longer valid, since it assumes constant specific
heats. In this case, Equation 2.79 is replaced by
ℎ1 = 𝑓 [ℎ6 + (𝑃i − 𝑃e )𝑣6 ] + (1 − 𝑓 )ℎi
(4.41)
and, of course, it is still true that if the pressure drop across the intake valves is neglected
𝑃 1 = 𝑃i
(4.42)
The volumetric efficiency and pumping work are
𝑒v =
𝑟(1 − 𝑓 )𝑣i
𝑚i
=
𝜌i 𝑉 𝑑
(𝑟 − 1)𝑣1
pmep = 𝑃e − 𝑃i
(4.43)
(4.44)
Finally, the net imep and thermal efficiency are
imepnet = imep − pmep
(4.45)
𝜂net = 𝜂 (1 − pmep∕imep)
(4.46)
The above equations are solved numerically in the four-stroke fuel--air Otto cycle
program FourStrokeFuelAir.m listed in Appendix F.14. The inputs to the four-stroke
114
Fuel--Air Combustion Processes
Otto fuel--air cycle program are the compression ratio 𝑟, the fuel--air equivalence ratio
𝜙, the intake pressure 𝑃i , the exhaust pressure 𝑃e , the intake temperature 𝑇i , and the
fuel type. Using the residual fraction (farg.m) and equilibrium combustion models
(ecp.m) developed in Chapter 3, it is possible to compute the properties at states 1, 2,
3, and 4. As in the four-stroke gas cycle, analysis of the four-stroke fuel--air cycle requires an additional iteration loop to determine the residual fraction and exhaust conditions.
EXAMPLE 4.7
Four-Stroke Fuel--Air Otto Cycle
Compute the cycle properties, volumetric efficiency, residual fraction, net imep, and net
thermal efficiency of a throttled four-stroke fuel--air Otto cycle with the following intake
conditions: gasoline fuel with 𝑃i = 50 kPa, 𝑃e = 105 kPa, 𝑇i = 300 K, 𝜙 = 0.8, and a
compression ratio of 10.
SOLUTION The four-stroke fuel--air Otto cycle program FourStrokeFuelAir.m is used to
compute the desired cycle parameters. The code computes the mixture properties at the
four cycle states, as well as the volumetric efficiency, residual fraction, net imep, and net
thermal efficiency. The input parameters are entered into the program as shown below:
Program for four-stroke fuel air cycle
establish initial conditions for intake stroke
clear;
Ti = 300;
% intake temperature (K)
Pi = 52.5;
% intake pressure (kPa)
Pe = 105;
% exhaust pressure (kPa)
phi = .8;
% equivalence ratio
rc = 10.;
% compression ratio
...
The program output is
FourStrokeFuelAir Results
Inlet: Temp (K)= 300.0
phi=
0.80
fuel=
State
Pressure (kPa)=
52.5
2
----
Pressure (kPa) =
1
---- 2
----
3 ----
4
52.5
1128.2
4138.1
237.3
348.5
748.9
2615.3
1507.3
Enthalpy (kJ/kgK) =
-196.5
254.5
810.8
-885.7
Int. Energy (kJ/kg =
-293.6
45.8
45.4
-1324.6
Temperature (K) =
Volume (m3/kg) =
1.850
0.185
0.185
1.849
Entropy (kJ/kgK) =
7.163
7.163
8.702
8.702
Cp (kJ/kg K) =
1.059
1.199
1.903
1.366
Work (kJ/kg) =
1031.0
Volumetric Efficiency =
0.9037
115
Four-Stroke Fuel--Air Otto Cycle
Ideal Thermal Efficiency =
Net Thermal Efficiency =
Imep (kPa) =
619.4
Pmep (kPa) =
52.5
0.491
0.4497
Exhaust Temperature (K) =
1262.6
Residual Fraction f = 0.0528
Note that the entropy is constant during compression and expansion, and the internal
energy and specific volume are constant during combustion. The temperature rise of the
inlet fuel--air mixture is about 48 K when mixed with the 𝑓 = 0.053 residual fraction. The
maximum temperature and pressure are 𝑇3 = 2615 K and 𝑃3 = 4138 kPa. The exhaust
temperature 𝑇e is 1507 K. The volumetric efficiency is 0.904, net imep is 619 kPa, and the
net thermal efficiency 𝜂net is 0.45.
(Imepnet /Pe)
Net indicated thermal efficiency
Results obtained from the program by varying the intake to exhaust pressure ratio
and the compression ratio are given in Figures 4.7 and 4.8. The net efficiency and the net
indicated mean effective pressure are each seen to be a strong function of the intake/exhaust
pressure ratio. The advantage of turbocharging and the disadvantage of throttling are clear.
For pressure ratios corresponding to supercharging, the curves are not representative, for
one would have to also account for the work to drive the compressor.
Figure 4.7 Effect of intake/exhaust
pressure ratio on four-stroke Otto
fuel--air cycle imep and thermal
efficiency.
r
10
8
0.5
0.4
0.3
0.2
C7H17, gasoline
= 0.8
Pe = 1.05 bar
Ti = 300 K
0.1
0
20
18
16
14
12
10
8
6
4
2
0
0.5
1.0
Intake exhaust pressure ratio (Pi /Pe)
r
10
8
1.5
Fuel--Air Combustion Processes
Volumetric efficiency, ev
116
1.2
1.1
r
1.0
8
10
0.9
0.8
C7H17 , gasoline
= 0.8
Pe = 1.05 bar
T i = 300 K
Residual fraction ( f )
0.10
Figure 4.8 Effect of intake/exhaust
pressure ratio on four-stroke Otto
fuel--air cycle residual fraction and
volumetric efficiency.
0.05
r
8
10
0
0.5
1.0
1.5
Intake to exhaust pressure ratio (Pi /Pe)
Notice that throttling also hurts the volumetric efficiency, mainly because of an increase
in the residual fraction. The residual fraction decreases with increasing compression ratio,
as one would expect.
The modeling of the intake and exhaust portion of the Otto cycle is not nearly as realistic
as the compression, combustion, and expansion portion of the cycle. This is because of
the assumptions of isobaric intake and exhaust processes and the neglect of heat transfer.
Neglect of the heat transfer causes the residual fraction to be underpredicted by a factor on
the order of 2. In Chapter 5, it will be shown that the processes are isobaric only at very low
piston speeds; consequently, at high piston speeds, the pumping mean effective pressure
can be in considerable error and can even have the wrong sign for super- or turbocharged
engines.
4.6 HOMOGENEOUS TWO-ZONE FINITE HEAT RELEASE CYCLE
In this section, we analyze a homogeneous fuel--air cycle in which the fuel and air are fully
mixed prior to the onset of combustion. We divide the combustion chamber into two zones,
burned (𝑏) and unburned (𝑢), and develop differential equations for the change in pressure
and change in temperature in each zone. The modeling is based on equations for energy and
mass conservation, equation of state, and mass fraction burned. The assumption is made
that both zones are at the same pressure 𝑃 , and the ignition temperature is the adiabatic
flame temperature based on the mixture enthalpy at the onset of combustion. The analysis
includes heat loss to the combustion chamber surfaces, and the blowby mass loss past
the rings.
In Chapter 2, we used a finite heat release function to express the fraction of heat added
over a given crank angle change. In this section, we use a similar function to represent
Homogeneous Two-Zone Finite Heat Release Cycle
117
the mass fraction 𝑥 of the cylinder contents that have burned. The solution procedure is
a simultaneous integration of a set of ordinary differential equations for 𝑃 , 𝑇u , 𝑇b , and
subsequent calculation of net work 𝑊 , heat loss 𝑄l , thermal efficiency 𝜂, and the indicated
mean effective pressure. This two-zone model has been extended to multi-zone models by
Raine et al. (1995).
The open system energy equation applied to the cylinder contents, is
𝑑𝑄
𝑑𝑉
𝑑𝑈 𝑚̇ l ℎl
−𝑃
=
+
𝑑𝜃
𝑑𝜃
𝑑𝜃
𝜔
=𝑚
𝑑𝑢
𝑑𝑚 𝑚̇ l ℎl
+𝑢
+
𝑑𝜃
𝑑𝜃
𝜔
(4.47)
The specific volume 𝑣 of the system is given by
𝑣=
𝑉
= 𝑥𝑣b + (1 − 𝑥)𝑣u
𝑚
(4.48)
Since 𝑣 = 𝑣(𝑇 , 𝑃 ), we can apply the chain rule to both zones:
𝜕𝑣 𝑑𝑇
𝜕𝑣 𝑑𝑃
𝜕𝑣b
= b b+ b
𝜕𝜃
𝜕𝑇b 𝑑𝜃
𝜕𝑃 𝑑𝜃
(4.49)
𝜕𝑣 𝑑𝑇
𝜕𝑣 𝑑𝑃
𝜕𝑣u
= u u+ u
𝜕𝜃
𝜕𝑇u 𝑑𝜃
𝜕𝑃 𝑑𝜃
(4.50)
Differentiating the equation for the specific volume, Equation 4.48, and incorporating
Equations 4.49 and 4.50 yield
𝑑𝑣
𝑑𝑣
𝑑𝑥
𝑉 𝑑𝑚
1 𝑑𝑉
−
= 𝑥 b + (1 − 𝑥) u + (𝑣b − 𝑣u )
𝑚 𝑑𝜃
𝑑𝜃
𝑑𝜃
𝑑𝜃
𝑚2 𝑑𝜃
𝜕𝑣 𝑑𝑇
𝜕𝑣 𝑑𝑇
1 𝑑𝑉
𝑉𝐶
+
= 𝑥 b b + (1 − 𝑥) u u
𝑚 𝑑𝜃
𝑚𝜔
𝜕𝑇b 𝑑𝜃
𝜕𝑇u 𝑑𝜃
[
]
𝜕𝑣b
𝜕𝑣u 𝑑𝑃
𝑑𝑥
+ 𝑥
+ (1 − 𝑥)
+ (𝑣b − 𝑣u )
𝜕𝑃
𝜕𝑃 𝑑𝜃
𝑑𝜃
(4.51)
(4.52)
The total internal energy 𝑢 of the system is the sum of the internal energy of burned
and unburned zones:
𝑈
(4.53)
= 𝑥𝑢b + (1 − 𝑥)𝑢u
𝑢=
𝑚
where 𝑢b is the internal energy of the burned gas at temperature 𝑇b , and 𝑢u is the energy of
the unburned gas at temperature 𝑇u . Since 𝑢 = 𝑢(𝑇 , 𝑃 ), we can again apply the chain rule
to both zones:
𝜕𝑢 𝑑𝑇
𝜕𝑢 𝑑𝑃
𝜕𝑢b
= b b+ b
𝜕𝜃
𝜕𝑇b 𝑑𝜃
𝜕𝑃 𝑑𝜃
(
)
)
(
(4.54)
𝜕𝑣b
𝜕𝑣b 𝑑𝑇b
𝜕𝑣b 𝑑𝑃
+𝑃
− 𝑇b
= 𝑐pb − 𝑃
𝜕𝑇b 𝑑𝜃
𝜕𝑇b
𝜕𝑃
𝑑𝜃
similiarly,
𝜕𝑢u
=
𝜕𝜃
(
(
)
)
𝜕𝑣u
𝜕𝑣u 𝑑𝑇u
𝜕𝑣u 𝑑𝑃
+𝑃
𝑐pu − 𝑃
− 𝑇u
𝜕𝑇u 𝑑𝜃
𝜕𝑇u
𝜕𝑃
𝑑𝜃
(4.55)
118
Fuel--Air Combustion Processes
𝜕𝑣
𝜕𝑣
Recall that the partial derivative terms 𝜕𝑇
and 𝜕𝑃
are computed by the programs ECP and
FARG for given mixture states.
The 𝑚𝑑𝑢∕𝑑𝜃 term in the energy equation is therefore
[
]
𝑑𝑢
𝑑𝑢
𝑑𝑥
𝑑𝑢
= 𝑚 𝑥 b + (1 − 𝑥) u + (𝑢b − 𝑢u )
𝑚
𝑑𝜃
𝑑𝜃
𝑑𝜃
𝑑𝜃
𝜕𝑣 𝑑𝑇
𝜕𝑣b 𝑑𝑇b
)
+ 𝑚(1 − 𝑥)(𝑐pu − 𝑃 u ) u
𝜕𝑇b 𝑑𝜃
𝜕𝑇u 𝑑𝜃
)
(
)]
[ (
(4.56)
𝜕𝑣b
𝜕𝑣u
𝜕𝑣b
𝜕𝑣u
𝑑𝑃
+𝑃
+𝑃
+ 𝑚(1 − 𝑥) 𝑇u
− 𝑚𝑥 𝑇b
𝜕𝑇b
𝜕𝑃
𝜕𝑇u
𝜕𝑃
𝑑𝜃
= 𝑚𝑥(𝑐pb − 𝑃
𝑑𝑥
𝑑𝜃
The term 𝑢𝑑𝑚∕𝑑𝜃 is the blowby term. As we modeled blowby in Chapter 2, we write
+ 𝑚(𝑢b − 𝑢u )
𝑚̇
𝑑𝑚
−𝐶𝑚
=− l =
(4.57)
𝑑𝜃
𝜔
𝜔
where C is the blowby coefficient depending on the ring design. This implies that the mass
in the cylinder at a given crank angle 𝜃, decreases as
𝑚(𝜃) = 𝑚1 exp[−𝐶(𝜃 − 𝜃1 )∕𝜔]
(4.58)
where 𝑚1 is the initial mass at state 1, the start of compression. The term 𝑑𝑄∕𝑑𝜃 is the heat
loss term, and as modeled in Chapter 2,
−𝑄̇ b − 𝑄̇ u
𝑄̇
𝑑𝑄
=− l =
(4.59)
𝑑𝜃
𝜔
𝜔
from the burned and unburned gases. The heat loss terms are expressed with a convection
equation,
𝑄̇ b = 𝐡𝐴b (𝑇b − 𝑇w )
𝑄̇ u = 𝐡𝐴u (𝑇u − 𝑇w )
(4.60)
where 𝐡 is the convection heat transfer coefficient, and 𝐴b and 𝐴u are the areas of the
burned and unburned gases in contact with the cylinder walls at temperature 𝑇w . We have
assumed, for convenience, that 𝐡u = 𝐡b = 𝐡 = constant. For the areas 𝐴b and 𝐴u , let us
suppose that the cylinder area 𝐴c can be divided as follows:
𝐴c =
𝜋𝑏2 4𝑉
+
2
𝑏
𝐴b = 𝐴c 𝑥1∕2
𝐴u = 𝐴c (1 − 𝑥
(4.61)
1∕2
)
The fraction of cylinder area contacted by the burned gas is assumed to be proportional
to the square root of the mass fraction burned to reflect the fact, because of the density
difference between burned and unburned gas, the burned gas occupies a larger volume
fraction of the cylinder than the unburned gas. In practice, the exponent on 𝑥 may be left
as a free parameter to be determined from experiments or a more complicated scheme may
be used based on an assumption about the flame shape.
We need to specify ℎl , the enthalpy of the mass loss due to blowby. Early in the
combustion process, unburned gas leaks past the rings. Late in the combustion process,
Homogeneous Two-Zone Finite Heat Release Cycle
119
burned gas leaks past the rings. Since a larger portion of unburned gas will be leaking than
the unburned mass fraction, let us assume that
ℎl = (1 − 𝑥2 )ℎu + 𝑥2 ℎb
(4.62)
As discussed in Chapter 3, the enthalpies ℎu = ℎ(𝑇u , 𝑃 ) and ℎb = ℎ(𝑇b , 𝑃 ) are computed
by the program ecp.m.
The mass fraction burned, 𝑥(𝜃), is represented by the following finite heat release
equation:
𝑥=0
𝜃 < 𝜃s
(
(
))
𝜋(𝜃 − 𝜃s )
1
𝑥=
1 − cos
2
𝜃b
𝑥=1
(4.63)
𝜃 > 𝜃s + 𝜃b
The remaining equation comes from introduction of the unburned gas entropy into
the analysis. Treating the unburned gas as an open system losing mass via leakage and
combustion, it can be shown that
−𝑄̇ u = 𝜔𝑚(1 − 𝑥)𝑇u
𝑑𝑠u
𝑑𝜃
(4.64)
Since 𝑠u = 𝑠u (𝑇u , 𝑃 ) it follows that
𝜕𝑠u
𝜕𝑠 𝑑𝑇
𝜕𝑠 𝑑𝑃
= u u+ u
𝜕𝜃
𝜕𝑇u 𝑑𝜃
𝜕𝑃 𝑑𝜃
=
𝑐pu 𝑑𝑇u
𝑇u 𝑑𝜃
+
𝜕𝑣u 𝑑𝑃
𝜕𝑇u 𝑑𝜃
(4.65)
Elimination of 𝑑𝑠u ∕𝑑𝜃 between Equations 4.64 and 4.65 gives
𝑐pu
𝑑𝑇u
𝜕𝑣 𝑑𝑃
−𝐡𝐴u
− 𝑇u u
=
(𝑇 − 𝑇w )
𝑑𝜃
𝜕𝑇u 𝑑𝜃
𝜔𝑚(1 − 𝑥) u
For convenience, let us define the following variables:
(
)
1 𝑑𝑉
𝑉𝐶
𝐴=
+
𝑚 𝑑𝜃
𝜔
]
[
𝐡𝐴c 1 𝜕𝑣b 1∕2
1 𝜕𝑣u
𝐵=
𝑥 (𝑇b − 𝑇w ) +
(1 − 𝑥1∕2 )(𝑇u − 𝑇w )
𝜔𝑚 𝑐pb 𝜕𝑇b
𝑐pu 𝜕𝑇u
]
[
𝑑𝑥 𝜕𝑣b ℎu − ℎb 𝑑𝑥 (𝑥 − 𝑥2 )𝐶
𝐶 = −(𝑣b − 𝑣u )
−
−
𝑑𝜃 𝜕𝑇b 𝑐pb
𝑑𝜃
𝜔
]
[
(
)
𝑇b 𝜕𝑣b 2 𝜕𝑣b
+
𝐷=𝑥
𝑐pb 𝜕𝑇b
𝜕𝑃
[
]
(
)
𝑇u 𝜕𝑣u 2 𝜕𝑣u
𝐸 = (1 − 𝑥)
+
𝑐pu 𝜕𝑇u
𝜕𝑃
The six equations to be integrated are
𝑑𝑃
𝐴+𝐵+𝐶
=
𝑑𝜃
𝐷+𝐸
(4.66)
(4.67)
120
Fuel--Air Combustion Processes
]
𝑑𝑇b
−𝐡𝐴c (𝑇b − 𝑇w ) 𝑇b 𝜕𝑣b 𝐴 + 𝐵 + 𝐶 ℎu − ℎb [ 𝑑𝑥
𝐶
+
=
+
− (𝑥 − 𝑥2 )
𝑑𝜃
𝑐pb 𝜕𝑇b 𝐷 + 𝐸
𝑥𝑐pb
𝑑𝜃
𝜔
𝜔𝑚𝑐pb 𝑥1∕2
𝑑𝑇u
−𝐡𝐴c (1 − 𝑥1∕2 )(𝑇u − 𝑇w ) 𝑇u 𝜕𝑣u 𝐴 + 𝐵 + 𝐶
=
+
𝑑𝜃
𝜔𝑚𝑐pu (1 − 𝑥)
𝑐pu 𝜕𝑇u 𝐷 + 𝐸
(4.68)
𝑑𝑉
𝑑𝑊
=𝑃
𝑑𝜃
𝑑𝜃
]
𝐡𝐴c [ 1∕2
𝑑𝑄l
𝑥 (𝑇b − 𝑇u ) + (1 − 𝑥1∕2 )(𝑇u − 𝑇w )
=
𝑑𝜃
𝜔
]
𝑑𝐻l
𝐶𝑚 [
(1 − 𝑥2 )ℎu + 𝑥2 ℎb
=
𝑑𝜃
𝜔
The above equations are solved numerically in the homogenous two-zone finite heat
release program Homogeneous.m listed in Appendix F.15. The inputs to the program
are the compression ratio 𝑟, engine bore and stroke, engine speed, heat transfer and blowby
coefficients, the fuel--air equivalence ratio 𝜙, residual fraction, the initial pressure 𝑃1 , and
temperature 𝑇1 . The initial burned gas temperature is assumed to be the adiabatic flame
temperature based on the enthalpy at the time of spark. If 𝑥 < 0.001, the system is treated as
consisting of only unburned gas, and if 𝑥 > 0.999, it is treated as being entirely composed
of burned gas.
Using the residual fraction (farg.m) and equilibrium combustion models (ecp.m)
developed in Chapter 3 to compute mixture mole fractions and properties, the Homogeneous.m program computes pressure, burned and unburned zone temperatures, work, and
cumulative heat and mass loss as a function of crank angle. Finally, the program computes
the net thermal efficiency of this cycle using Equation 4.69.
𝜂=
EXAMPLE 4.8
𝑤c.v. (1 + 𝜙𝐹 𝐴s )
(1 − 𝑓 )𝜙𝐹 𝐴s 𝑎o
(4.69)
Homogeneous Two Zone Finite Heat Release
Compute the pressure and burned and unburned zone temperatures, net imep, and net
thermal efficiency of a homogeneous finite heat release cycle with the following initial
conditions: gasoline fuel with 𝑃1 = 100 kPa, 𝑇1 = 350 K, and 𝜙 = 0.8.
The engine has a bore of 0.10 m and stroke of 0.10 m, with a half stroke to rod
ratio 𝜖 = 0.25, a compression ratio 𝑟 = 10, residual fraction 𝑓 = 0.1, and operates at
𝑁 = 2000 rpm. The start of heat release is at −35◦ , and the combustion duration is
60◦ . Assume the cylinder wall temperature is 420 K, with a heat transfer coefficient
of 500 W/(m2 K), and blowby coefficient of 0.8 𝑠−1 .
SOLUTION The input parameters are entered into the Homogeneous.m program as shown
below:
Homogeneous Two Zone Combustion Cycle
This program computes the pressure and temperature
vs crank angle,
the work, indicated thermal efficiency,
and the Indicated mean effective pressure (kPa)
R = 10;
Compression ratio
Homogeneous Two-Zone Finite Heat Release Cycle
B = .10;
Bore - B (m)
S = .08;
Stroke - S (m)
EPS = 0.25;
Half stroke to rod ratio
RPM = 2000;
Engine speed (RPM)
HEAT = 500;
Heat transfer coefficient (W/m2-K)
121
BLOWBY = 0.8; Blowby coefficient
THETAB = 60;
Burn angle (Deg)
THETAS = -35;
Start of heat release (deg ATDC)
PHI = 0.8;
Equivalence ratio
F = 0.1;
Residual fraction
TW = 420;
Wall temperature (K)
fuel_id = 2;
gasoline
FS = 0.06548;
gasoline stoichiometric fuel{\ndash}air ratio
A0 = 47870;
maximum available energy (kJ/kg)
T1 = 350;
Initial temperature (K)
P1 = 100;
Initial Pressure (kPa)
...
The program output is shown in Figures 4.9--4.12, which are respectively plots of heat
release fraction, pressure, unburned and burned temperatures, and cumulative work and
heat loss as a function of crank angle. The burn fraction 𝑥 begins at −35◦ , and ends at +25◦ .
The maximum pressure is about 6000 kPa at a crank angle of +15◦ atdc. The unburned
gas temperature 𝑇u profile rises due to the compression process and heat transfer from the
cylinder walls, and ends at +25◦ , as combustion is completed. The burned gas temperature
𝑇b begins at −35◦ at the adiabatic flame temperature = 2140 K, as heat release is initiated,
increases to about 2500 K, then decreases to about 1100 K as the cylinder volume increases.
The cumulative work is initially negative during compression, then becomes positive
during expansion. The heat loss is very small during compression, then increases as the
cylinder temperature increases during combustion. The mass loss is linear, due to the simple
blowby model used.
1
Burn fraction
0.8
0.6
0.4
0.2
0
Figure 4.9 Heat release fraction
versus crank angle.
–50
0
Crank angle
50
Fuel--Air Combustion Processes
7000
Pressure (kPa)
6000
5000
4000
3000
2000
1000
0
–100
Figure 4.10 Pressure versus crank
angle.
–50
0
Crank angle
Figure 4.11 Unburned and burned
zone temperature versus crank angle.
50
100
50
100
Crank angle
600
Work
Heat loss
500
Work and heat loss (J)
122
400
300
200
100
0
–100
–200
–300
Figure 4.12 Work and heat loss (J)
versus crank angle.
–100
–50
0
Crank angle
The net thermal efficiency, from Equation 4.69, is 0.388, and the net imep is 950 kPa.
The equivalent fuel--air Otto cycle predicts 𝜂 = 0.460, hence 𝜂∕𝜂Otto = 0.845.
Comparison of Fuel--Air Cycles with Actual Spark Ignition Cycles
123
4.7 COMPARISON OF FUEL--AIR CYCLES WITH
ACTUAL SPARK IGNITION CYCLES
Since the efficiency of an actual engine must be less than the efficiency of its equivalent Otto
fuel--air cycle, the fuel--air cycle is a convenient reference for comparison. The indicated
efficiency and mean effective pressure of actual engines are determined in practice by
measuring the cylinder pressure as a function of cylinder volume and integrating ∫ 𝑃 𝑑𝑉 to
find the work. It is also possible to measure the residual fraction and charge density trapped
within the cylinder.
With reference to Figure 4.13 an equivalent fuel--air cycle is constructed by matching
the temperature, pressure, and composition (and thereby entropy) at some reference point
after closing of the intake valve and prior to firing of the spark plug.
Since the actual process is nearly isentropic, the compression curves of the two cycles
nearly coincide. Soon after the onset of combustion, the actual cycle pressure starts rising
above that of the fuel--air cycle. Because the combustion actually is not at constant volume,
the peak pressure is considerably less than that predicted by the fuel--air cycle. The expansion curve 3--4 is polytropic in character; measurements show that the entropy decreases
during expansion, primarily due to heat transfer to the coolant. At point 4 the exhaust valve
opens, and soon after, the pressure falls rapidly to the exhaust pressure. The cross-hatched
area represents ‘‘lost work’’ that can mainly be attributed to the following:
•
•
•
•
Heat loss
Mass loss
Finite burn rate
Finite blowdown rate
By inspection of Tables 4.4--4.7, a series of CFR engine data sets from Taylor (1985),
compared with equivalent fuel--air Otto cycle predictions, the following conclusions can
be drawn:
Pressure
1. The indicated mean effective pressure is maximized slightly rich of stoichiometric, and
increases with increasing compression ratio and inlet pressure.
3
0 Intake valve closes
1 Reference point
2 Spark fires
3 Combustion ends
4 Exhaust valve opens
Fuel–air cycle
Actual cycle
Lost work
2
Figure 4.13 Comparison of an actual
spark ignition cycle with its equivalent
fuel--air cycle. Adapted from Taylor
(1985).
4
1
Volume
0
124
Fuel--Air Combustion Processes
Table 4.4 Effect of Equivalence Ratio 𝜙 on CFR Engine Performance
𝜙
𝜃s (atdc)
(deg)
0.74
0.80
1.17
1.80
−33
−23
−15
−20
𝜃𝑑
(deg)
imep
(bar)
𝜂∕𝜂Otto
58
39
33
39
6.3
6.4
7.7
7.0
0.85
0.83
0.85
0.83
𝑁 = 1200 rpm, 𝑟 = 7,
Source: Taylor (1985).
Table 4.5 Effect of Spark Advance 𝜃s on CFR Engine Performance
𝜃s (atdc)
(deg)
𝜃𝑑
(deg)
bmep
(bar)
imep
(bar)
𝜂∕𝜂Otto
40
40
38
39
5.0
5.7
5.8
5.0
6.0
7.5
7.5
6.9
0.73
0.82
0.82
0.74
0
−13
−26
−39
𝑁 = 1200 rpm, 𝑟 = 6, 𝜙 = 1.13,
Source: Taylor (1985).
Table 4.6 Effect of Engine Speed 𝑁 on CFR Engine Performance
𝑁
(rpm)
𝜃𝑠 (atdc)
(deg)
𝜃𝑑
(deg)
bmep
(bar)
imep
(bar)
𝜂∕𝜂Otto
900
1200
1500
1800
−18
−19
−22
−18
36
39
40
38
3.90
3.77
3.80
3.59
5.89
5.94
6.07
6.14
0.842
0.848
0.865
0.877
𝑟 = 6, 𝜙 = 1.13,
Source: Taylor (1985).
Table 4.7 Effect of Compression Ratio 𝑟 on CFR Engine Performance
𝑟
𝜃s (atdc)
(deg)
𝜃𝑑
(deg)
bmep
(bar)
imep
(bar)
𝜂∕𝜂Otto
8
7
6
5
4
−13
−14
−15
−16
−17
29
31
33
37
39
5.5
5.3
5.3
4.8
4.1
7.9
7.9
7.2
6.8
6.1
0.79
0.86
0.84
0.87
0.86
𝑁 = 1200 rpm, 𝜙 = 1.13,
Source: Taylor (1985).
Limited Pressure Fuel--Air Cycle
125
2. The ratio of the actual efficiency to the equivalent fuel--air Otto cycle 𝜂∕𝜂Otto is on
the order of 0.85, and varies insignificantly with engine operating variables, at most
decreasing slightly with increasing compression ratio.
3. The combustion duration, 𝜃𝑑 , is on the order of 35◦ , decreases with increasing compression ratio or inlet pressure, and is minimum at a slightly rich equivalence ratio.
4. The optimum spark advance, 𝜃s , increases with combustion duration and with increased
engine speed. The optimum spark advance is defined as the crank angle 𝜃 that produces
maximum brake torque (MBT).
5. The imep increases with engine speed, while bmep decreases, which, as we will see in
Chapter 10, is caused by increased friction.
Based on the analysis done in Chapter 2, these results are to be expected. The differences
between the fuel--air model and the actual engine is primarily due to heat loss, but also to
mass loss and the finite burning rate. A small part of the discrepancy can be also attributed
to opening the exhaust valve prior to bottom dead center to provide for the finite flowrate
of the blowdown process.
For a given engine operated at optimum spark timing, the ratio 𝜂∕𝜂Otto is nearly
independent of the fuel--air equivalence ratio, the inlet temperature, the inlet pressure, the
exhaust gas recirculation, and the engine speed. All the trends predicted by the Otto fuel--air
cycle are, in fact, observed in practice.
This implies that there is slightly greater potential for improving the efficiency of
spark ignition engines by increasing their theoretical efficiency through an increase in
compression ratio rather than by reducing their losses. To illustrate, suppose that by reducing
the heat loss or increasing the burn rate one could increase 𝜂∕𝜂Otto from 0.80 to 0.90. The
efficiency might be 0.32 instead of 0.29. On the other hand, suppose that research results
showed that the compression ratio could be increased to 20. The fuel--air cycle efficiency
would increase to about 0.46, and if 𝜂∕𝜂Otto were still 0.8, the actual efficiency would
now be 0.37. There is greater potential with this approach because the second law of
thermodynamics does not limit the choice of variables that fix the theoretical efficiency
but it does limit the gains that can be realized once the parameters that specify the fuel--air
cycle are fixed.
4.8 LIMITED PRESSURE FUEL--AIR CYCLE
This cycle developed in this section models diesel engines and fuel-injected stratified
charge engines in which the fuel is injected at the time it is intended to burn. The processes
are
1 to 2
2 to 2.5
2.5 to 3
3 to 4
Isentropic compression of air and residual gas
Constant volume, adiabatic fuel injection, and combustion
Constant pressure, adiabatic fuel injection, and/or combustion
Isentropic expansion
These engines, in general, are fueled overall lean. In this case, the air--residual gas
mixture is equivalent to equilibrium combustion products at an equivalence ratio given by
Equation 4.70:
𝜙12 =
𝑓𝜙
1 + (1 − 𝑓 )𝜙𝐹s
(4.70)
126
Fuel--Air Combustion Processes
where the residual fraction, 𝑓 , is the ratio of the residual mass to the cylinder gas mass prior
to fuel injection. The thermodynamic state during compression can then be determined with
the equivalence ratio 𝜙 used as an argument.
The details of the fuel injection and combustion are of no concern at this level of
modeling. We need only assume that at state 3 the gases in the cylinder are equilibrium
combustion products at the overall fuel--air equivalence ratio. To specify the state, we know
that
𝑃3 ≤ 𝑃limit
(4.71)
and we apply the energy equation to the process 2 to 3:
Δ𝑈 = 𝑚3 𝑢3 − 𝑚2 𝑢2 = 𝑚f ℎf − 𝑃3 (𝑚3 𝑣3 − 𝑚3 𝑣2 )
(4.72)
𝑚𝑎 + 𝑚r
𝑚2
1
=
=
𝑚3
𝑚𝑎 + 𝑚 r + 𝑚f
1 + (1 − 𝑓 )𝜙𝐹 𝐴s
(4.73)
𝑚 − 𝑚2
(1 − 𝑓 )𝜙𝐹 𝐴s
𝑚f
= 3
=
𝑚3
𝑚3
1 + (1 − 𝑓 )𝜙𝐹 𝐴s
(4.74)
It follows that
and the enthalpy at state 3 is
ℎ3 = 𝑢3 + 𝑃3 𝑣3 =
𝑢2 + 𝑃3 𝑣2 + (1 − 𝑓 )𝜙 𝐹 𝐴s ℎf
1 + (1 − 𝑓 )𝜙𝐹 𝐴s
(4.75)
The pressures during fuel injection 𝑃f are high enough that Equation 3.42 should be used
in lieu of Equation 3.47 to evaluate the fuel enthalpy. Hence,
ℎfuel = ℎof + 𝑣o (𝑃fuel − 𝑃o )
(4.76)
where the subscript zero denotes conditions at atmospheric pressure (𝑃o = 1.01325 bar).
In doing a computation, one should first assume that the combustion and fuel injection
are entirely at constant volume. If the resultant 𝑃3 satisfies Equation 4.71, then indeed the
process is at constant volume. However, if Equation 4.71 is not satisfied, then 𝑃3 = 𝑃limit ,
and one solves Equation 4.75 to find the state 3.
The expansion occurs to a specific volume at state 4 different from that at state 1
because of the fuel injected. It can be shown that expansion must satisfy the following
constraints:
𝑠4 = 𝑠3
𝑟𝑣2
𝑚2
𝑣4 = 𝑟𝑣3 = 𝑟𝑣2
=
𝑚3
1 + (1 − 𝑓 )𝜙𝐹 𝐴s
(4.77)
(4.78)
To evaluate the work 𝑤c.v. , it is convenient to split it into three parts, 𝑤12 , 𝑤23 , and 𝑤34 ,
due to the change in mass and energy by fuel injection. The work components expressed
per unit mass after fuel injection are
𝑤c.v. = 𝑤12 + 𝑤23 + 𝑤34
𝑢1 − 𝑢 2
𝑊12 𝑚2
=
𝑚2 𝑚 3
1 + (1 − 𝑓 )𝜙𝐹 𝐴s
]
[
𝑣2
= 𝑃3 𝑣3 −
1 + (1 − 𝑓 )𝜙𝐹 𝐴s
(4.79)
𝑤12 =
(4.80)
𝑤23
(4.81)
𝑤34 = 𝑢3 − 𝑢4
(4.82)
127
Limited Pressure Fuel--Air Cycle
Pl /P1
100
75
Thermal efficiency
0.8
0.7
50
0.6
Pl /P1
100
75
50
0.4
(Imep/P1)
0.3
Figure 4.14 Effect of equivalence ratio on
limited pressure fuel--air cycle.
0.2
20
18
15
16
13
14
12
10
8
6
4
2
0
Diesel fuel
r = 15
P1 = 1 bar
T1 = 325 K
f = 0.05
Minimum to give
Pl /P1 = 100
0.2
Pl /P1
100
75
50
0.4
0.6
0.8
1.0
Equivalence ratio ( )
The efficiency and imep are given by
𝜂=
𝑤 [1 + (1 − 𝑓 )𝜙𝐹 𝐴s ]
𝑤c.v. 𝑚3
= c.v.
𝑚f 𝑎 o
(1 − 𝑓 )𝜙𝐹 𝐴s 𝑎o
imep =
𝑤c.v. [1 + (1 − 𝑓 )𝜙𝐹 𝐴s ]
𝑣1 − 𝑣2
(4.83)
(4.84)
Results obtained using the above modeling for different compression and equivalence ratios
are given in Figures 4.14 and 4.15. Important conclusions are
1. The efficiency decreases with increased equivalence ratio. This is consistent with the
Otto cycle fuel--air model, where the efficiency is maximum for lean mixtures.
2. The imep increases with equivalence ratio. This is also consistent with the Otto cycle
fuel--air model, where the imep is maximum for slightly rich mixtures. Since the combustion is heterogeneous, the engine can be controlled without throttling the air.
3. The efficiency and imep are a weaker function of compression ratio relative to an Otto
cycle. This is due to the constraint on peak pressure.
4. Both efficiency and imep increase with increasing limit pressure.
5. Even in the absence of heat and mass loss, the ratio 𝜂∕𝜂Otto may be as low as 0.85.
Fuel--Air Combustion Processes
(Imep/P1)
Indicated thermal efficiency
128
Figure 4.15 Effect of compression ratio
on limited pressure fuel--air cycle.
0.8
Pl /P1
100
0.7
75
50
0.6
100
75
0.4
50
0.3
15
16
13
14
12
10
8
6
4
2
0
10
Diesel fuel
= 0.8
P1 = 1 bar
T1 = 325 K
f = 0.05
Minimum r giving
Pl /P1 = 100
100
75
50
Maximum r giving
Pl /P1 = 50
12
14
16
18
20
22
Compression ratio (r)
The constraint on peak pressure results in the efficiency and imep being insensitive
to compression ratio. In practice, the ratio 𝜂∕𝜂Otto for diesel and fuel injected stratified
engines is more sensitive to the particular design and the operating conditions than it is for
homogeneous charge spark ignition engines. Thus, a greater range of indicated efficiencies
exists among engines made by different manufacturers and among engines of different
sizes. Divided chamber engines usually have a smaller 𝜂∕𝜂Otto ratio than open chamber
engines partly because of throttling losses through the throat between chambers, but mainly
because of a greater heat loss.
4.9 COMPARISON OF LIMITED PRESSURE FUEL--AIR CYCLES
WITH ACTUAL COMPRESSION IGNITION CYCLES
Diesel engines are designed to limit both the rates of pressure rise and the maximum
pressures to satisfy durability, noise, and emissions considerations. Therefore, a convenient
standard appears to be the equivalent limited pressure fuel--air cycle, and indeed this was
the choice of Taylor (1985). As in the spark ignition engine, the losses are attributed to
heat and mass loss, the finite blowdown rate, and combustion occurring at less than the
maximum pressure.
The fuel--air cycle adequately models conventional spark ignition engines, but is not
as useful for an engine as heterogeneous as a typical diesel engine. Diesel engine fuel flow
rates are limited by the appearance of solid carbon in exhaust that did not burn to carbon
Homework
129
monoxide or carbon dioxide. This occurs even though the engine is running lean and is not
predicted by fuel--air cycles. A more sophisticated model is required. These exist but are
beyond the scope of this text.
There are two problems with using the limited pressure fuel--air cycle as a standard.
The first is that an engine that can operate at a higher peak pressure and still satisfy the
constraints imposed by durability, noise, and emissions considerations is a better engine
and ought to be recognized as such. The second issue is that for some engines, it is not
possible to construct an equivalent limited-pressure fuel--air cycle because the losses are so
great that the peak pressure is less than would be achieved via isentropic compression alone.
We conclude this chapter by noting that if the ratio of 𝜂∕𝜂Otto is a measure of how well
an engine of a given compression ratio is developed, it appears that gasoline engines are
more highly developed than diesel engines. This suggests that there is more potential for
payoff from research and development on losses in diesel engines than there is on losses in
spark ignition engines.
For further reading on the topic of how the second law of thermodynamics can be
used to better understand internal combustion engine processes, especially combustion, the
reader is referred to the text by Bejan (2006), and a series of papers by Caton (2000, 2010).
4.10 REFERENCES
BEJAN, A. (2006), Advanced Engineering Thermodynamics, Wiley, New York.
CATON, J. (2000), ‘‘On the Destruction of Availability (Exergy) Due to Combustion Processes: With
Specific Application to Internal Combustion Engines,’’ Energy, Vol. 25, pp. 1097--1117.
CATON, J. (2010), ‘‘An Assessment of the Thermodynamics Associated the High-Efficiency Engines,’’
ASME paper ICEF2010-35037.
OBERT, E. (1973), Internal Combustion Engines and Air Pollution, Harper & Row, New York.
RAINE, R., C. STONE, and J. GOULD (1995), ‘‘Modeling of Nitric Oxide Formation in Spark Ignition
Engines with a Multizone Burned Gas,’’ Combust. Flame, Vol. 102, p. 241--255.
TAYLOR, C. (1985), The Internal Combustion Engine in Theory and Practice, Vol. 1, MIT Press,
Cambridge, Massachusetts.
4.11 HOMEWORK
4.1
Compute the higher, lower, and equilibrium heats of combustion for methanol CH3 OH (𝑙).
The equilibrium computation determines the quality of the water in the products at standard
atmospheric pressure and temperature.
4.2
The heat of combustion could have been defined without requiring complete conversion to
carbon dioxide and water. What would the lower heat of combustion be for the case 𝜙 =
1.4, fuel = C8 H18 (𝑙) octane, 𝑇o = 298 K, 𝑃o = 1 atm? Assume the water quality 𝜒 = 1 and
that the equilibrium constant 𝐾 = 9.95 × 10−6 .
4.3
With reference to Figure 4.2, explain why the heats of combustion at 𝜙 = 0.2 and 𝜙 = 1.2
are less than those at 𝜙 = 1.0.
4.4
What is the residual mass fraction required to reduce the adiabatic flame temperature of
gasoline, diesel, methane, methanol, and nitromethane below 2000 K? Assume 𝜙 = 1.0 at
101 kPa and 298 K.
4.5
Plot the adiabatic flame temperature of gasoline as a function of pressure (50 < 𝑃 <5000
kPa) for T = 298 K, 𝜙 = 1.0, and 𝑓 = 0.1.
130
Fuel--Air Combustion Processes
4.6
Equilibrium combustion products at 𝜙 = 0.9 of methane CH4 are expanded isentropically
from 𝑇1 = 2000 K, 𝑃1 = 1000 kPa to a pressure 𝑃2 of 100 kPa. Find the final temperature
𝑇2 and the work done.
4.7
Equilibrium combustion products of gasoline are expanded isentropically by a volume ratio
of 10:1. (a) For 𝜙 = 1.1 and an initial state of 𝑇1 = 3000 K, 𝑃1 = 5000 kPa, find the final
state (𝑇2 , 𝑃2 ) and the work done. (b) Repeat the calculation for 𝜙 = 0.9. What is the effect
of equivalence ratio?
4.8
Show that Equation 4.35 for the second law efficiency reduces to Equation 4.26 for the
first law efficiency when there is only heat transfer to the system instead of combustion.
4.9
Derive Equation 4.37 for the second law efficiency.
4.10
Derive Equation 4.43 for the volumetric efficiency.
4.11
What is the change in exergy 𝑎𝑗 − 𝑎i (kJ/kg) for the compression (1--2), combustion (2--3),
and expansion (3--4) strokes of the fuel--air cycle of Example 4.3?
4.12
Compute 𝑎o , the maximum energy of combustion, for liquid gasoline C7 H17 based on
equilibrium water quality, lean combustion at 𝜙 = 0.01, and unmixed reactants.
4.13
Plot the second law thermal efficiency versus compression ratio (vary 𝑟 from 5 to 20) for
a methane fuel--air Otto cycle. Compare the results with the gas Otto cycle. Assume 𝑇i =
350 K, 𝜙 = 1.0, 𝑓 = 0.05, and 𝛾 = 1.3.
4.14
What compression ratio is required to have an imep of 1500 kPa with a methane fuel--air
Otto cycle, assuming 𝑇1 = 325 K, 𝜙 = 0.95, 𝑓 = 0.05, and 𝑃1 = 101.3 kPa?
4.15
What value of the equivalence ratio will maximize the imep for a gasoline fuel--air Otto
cycle with a compression ratio of 10? Assume 𝑇1 = 350 K, 𝑓 = 0.05, and 𝑃1 = 101.3 kPa.
4.16
Exhaust gas recirculation (EGR) is used in spark ignition engines to reduce the peak
combustion temperature and the concentration of NO𝑥 . EGR can be modeled in a fuel--air
cycle by varying the residual fraction. What residual fraction is needed to reduce 𝑇3 to
2250 K in a gasoline engine with the following conditions: 𝑇1 = 350 K, 𝜙 = 0.95, 𝑟 = 10,
and 𝑃1 = 101.3 kPa? If the original residual fraction was 𝑓 = 0.05, what is the change in
the imep, and why?
4.17
Plot and discuss the effect of supercharging on the volumetric and thermal efficiency of
a four-stroke gasoline fuel--air Otto cycle model. Vary the intake pressure from 60 to 160
kPa, and assume 𝑇i = 300 K, 𝜙 = 1.0, 𝑟 = 10, and 𝑃e = 105 kPa.
4.18
As the intake pressure is throttled from 101.3 to 50 kPa, what is the change in the volumetric
efficiency, imep, and residual fraction of a four-stroke gasoline fuel--air Otto cycle engine?
Assume 𝑇i = 300 K, 𝜙 = 0.9, 𝑟 = 11, and 𝑃e = 105 kPa.
4.19
A CFR engine is operated at the following conditions: gasoline fuel with 𝑃1 = 100 kPa, 𝑇1 =
340 K, 𝑟=10, 𝜙 = 1.0. The engine has a bore of 0.0825 m and stroke of 0.114 m. The other
engine conditions are the same as in Example 4.8. Using the program Homogeneous.m,
(a) compute the pressure and burned and unburned zone temperatures, imep, and thermal
efficiency. (b) While keeping other parameters constant, compute and plot the effect of
varying the spark advance (vary 𝜃s from 0 to −40◦ atdc), the equivalence ratio (vary 𝜙
from 0.7 to 1.3), and the compression ratio (vary 𝑟 from 5 to 15). Compare and discuss the
computational results with the experimental results given in Section 4.7.
4.20
Derive Equations 4.70--4.83 for the limited pressure fuel--air cycle.
Chapter
5
Intake and Exhaust Flow
5.1 INTRODUCTION
In this chapter, we examine the airflow into and out of the intake and exhaust systems
in internal combustion engines. We will use compressible fluid mechanics to develop
relationships between engine speed, mass flow rate, and valve geometry. A fundamental
limiting factor affecting the performance of internal combustion engines is the onset of
choked flow that occurs at high engine speeds. Since choked flow results from intake and
exhaust valve flow area restrictions, most present-day engines have multiple intake and
exhaust valves to minimize flow restrictions.
So far we have restricted our attention to constant pressure intake and exhaust flows.
The constant pressure intake and exhaust model is qualitatively useful but quantitatively
lacking, since no account is made of the unsteady and compressible flow phenomena
affecting mass flow and pressure. Finally, we discuss the performance of superchargers
and turbochargers, and how they can be incorporated into internal combustion engines to
increase airflow and power.
5.2 VALVE FLOW
Valve Flow and Discharge Coefficients
The most significant airflow restriction in an internal combustion engine is the flow through
the intake and exhaust valves. Typically the minimum cross-sectional area in the intake and
exhaust system occurs at the valve, as shown in Figures 5.1 and 5.2. In accounting for the
pressure drop across the intake and exhaust valves considerable success has been realized
by modeling the gas flow through the valves as one-dimensional quasi-steady compressible
flow. In this section, we obtain relationships for the mass flow rate through valves as a
function of the intake and exhaust pressure ratio.
The velocity 𝑈 in the intake and exhaust manifolds can be non dimensionalized by the
speed of sound 𝑐 to form the Mach number 𝑀 = 𝑈 ∕𝑐, and conditions at various locations
in the manifolds can be related to stagnation conditions where the Mach number 𝑀 = 0.
For steady adiabatic flow, the energy equation is
ℎo = ℎ +
𝑈2
= const
2
(5.1)
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
131
132
Intake and Exhaust Flow
A1
Pv = pressure at A1 or A2
Pc = cylinder pressure
l = valve lift
d = valve diameter
A2
d
Pc
Pv
Figure 5.1 Schematic
of valve flow areas.
l
dt
ds
di
Figure 5.2
blockage.
Schematic of valve flow
and for a gas with constant specific heats
𝑇o
𝑈2
= 1+
𝑇
2𝑐𝑝 𝑇
= 1+
(
𝛾 −1
2
)
(5.2)
𝑀2
For isentropic flow, the pressure and density are functions of the Mach number:
[ ]𝛾∕(𝛾−1)
𝑃o
𝑇
= o
𝑃
𝑇
[
(
)
]𝛾∕(𝛾−1)
𝛾 −1
= 1+
𝑀2
2
(5.3)
[ ]1∕(𝛾−1)
𝜌o
𝑇
= o
𝜌
𝑇
[
(
)
]1∕(𝛾−1)
𝛾 −1
2
= 1+
𝑀
2
(5.4)
Valve Flow
133
The pressure and density at the valve are related to the upstream stagnation pressure
and density by the isentropic relation
( )1∕𝛾
𝑃𝑣
(5.5)
𝜌𝑣 = 𝜌o
𝑃o
and the ideal gas equation at stagnation conditions is
𝑃o = 𝜌o 𝑅𝑇o
(5.6)
The stagnation sound speed 𝑐o is
𝑐o = (𝛾𝑅𝑇o )1∕2
(5.7)
The mass flow rate, 𝑚,
̇ through a valve depends on the valve effective area 𝐴f , fluid
velocity and density:
𝑚̇ = 𝜌v 𝐴f 𝑈is
(5.8)
The velocity 𝑈is is the reference isentropic velocity, and 𝜌v is the fluid density at the
valve. The isentropic velocity 𝑈is depends on the pressure ratio and is calculated from the
isentropic relation for flow in a converging nozzle:
[
(
( )(𝛾−1)∕𝛾 )]1∕2
𝑃v
𝛾 𝑃o
(5.9)
𝑈is = 2
1−
𝛾 − 1 𝜌o
𝑃o
where
𝑃o = upstream total or stagnation pressure
𝑃v = valve static pressure
𝜌o = upstream total or stagnation density
Upon substitution of Equations 5.5, 5.6, 5.7, and 5.9 into Equation 5.8, we obtain the
desired relationship:
(( )
[
( )(𝛾+1)∕𝛾 )]1∕2
𝑃v
𝑃v 2∕𝛾
2
𝑚̇ = 𝜌o 𝐴f 𝑐o
(5.10)
−
𝛾 −1
𝑃o
𝑃o
For intake flow into the cylinder, the stagnation conditions refer to conditions upstream
of the valve in the intake port. For exhaust flow out of the cylinder, the stagnation conditions
refer to conditions in the cylinder.
Choked flow occurs at a valve throat if the ratio of the upstream pressure to downstream
pressure exceeds a critical value. When the flow is choked the Mach number at the valve
throat is 𝑀 = 1, and the critical pressure ratio is found from Equation 5.3 above to be
[
(
) ]𝛾∕(𝛾−1)
𝑃o
𝛾 −1
= 1+
𝑃
2
(
=
𝛾 +1
2
)𝛾∕(𝛾−1)
(5.11)
For 𝛾 = 1.35, the critical pressure ratio is 1.86. Note that for choked flow, the valve
static pressure 𝑃v depends only on the upstream stagnation pressure 𝑃o and is independent
of the downstream pressure. For nonchoked flow into the cylinder, it may generally be
assumed that the throat pressure is equal to the cylinder pressure. If the kinetic energy in
134
Intake and Exhaust Flow
the cylinder is relatively negligible, one need not distinguish between static and stagnation
cylinder pressure. However, for exhaust flow from the cylinder in nonchoked situations,
one equates the throat pressure to the exhaust port static pressure and this may differ
significantly from the exhaust port stagnation pressure.
Upon substitution of Equation 5.11 into Equation 5.10, we find the choked mass flow
rate 𝑚̇ cr to be
)(𝛾+1)∕2(𝛾−1)
(
2
𝑚̇ cr = 𝜌o 𝐴f 𝑐o
𝛾 +1
𝑃
= 𝐾(𝛾)𝐴f √ o
𝑅𝑇o
(5.12)
where 𝐾(𝛾) is a parameter dependent only on the specific heat ratio 𝛾. For 𝛾 = 1.35, 𝐾 =
6.76.
)(𝛾+1)∕2(𝛾−1)
(
2
1∕2
(5.13)
𝐾(𝛾) = 𝛾
𝛾 +1
Alternatively, the effective area required for a given mass flow rate and stagnation
pressure and temperature is
√
𝑚̇ cr 𝑅𝑇o
𝐴f =
(5.14)
𝐾𝑃o
Equation 5.10 for the mass flow rate assumes flow from an upstream reservoir through
an effective minimum valve area, 𝐴f . The effective valve area depends on the valve
diameter and lift, and two associated minimum areas are used, each with a corresponding
flow coefficient. As shown in Figure 5.1, a geometric minimum area 𝐴v can be defined
using either the valve curtain area 𝐴1 = 𝜋𝑑𝑙 or the valve seat area 𝐴2 = 𝜋𝑑 2 ∕4. If the valve
seat area is chosen, the flow coefficient is labeled 𝐶f , as defined in Equation 5.15. If the
valve curtain area is used, the flow coefficient is labeled 𝐶d , a discharge coefficient, as
shown in Equation 5.16.
𝜋
(Seat)
(5.15)
𝐴 f = 𝐶 f 𝐴 v = 𝐶f 𝑑 2
4
𝐴f = 𝐶d 𝐴v = 𝐶d 𝜋𝑑𝑙
(Curtain)
(5.16)
In the idealized model of a poppet valve shown in Figure 5.1, the two minimum
geometric area 𝐴v possibilities are evident, depending on the valve lift. For low lift the
minimum area is the valve curtain area, and for larger lifts the minimum area is the valve
seat area. In this idealized model, the geometric effects of the valve stem and valve seat
angle are neglected. These considerations are addressed in Homework Problem 7.7. As
shown in Figure 5.3, there is little reason to open a valve much beyond 𝑙∕𝑑 ≈ 1∕4, since
the flow area at such lifts would be limited by the port size. For intake ports, the maximum
𝑙∕𝑑 is about 0.4, accounting for the flow coefficient of the port.
Flow or discharge coefficients are measured using steady flow benches like that illustrated in Figure 5.4. The mass flow rate and pressure drop across the valve are measured
for a number of different valve lifts and pressure ratios. Equation 5.10 is then solved for the
flow coefficient for a particular choice of representative valve area. It should be noted that
flow bench pressure drops are of the order of 5 kPa, whereas actual pressure drops across
exhaust valves are about two orders of magnitude larger, since the cylinder pressure at the
exhaust valve opening is of the order of 500 kPa.
A typical plot of 𝐶f versus lift is given in Figure 5.5. The flow coefficient 𝐶f increases
monotonically from zero with lift, since the effective flow area through the valve increases
Valve Flow
135
Geometric
1.00
0.75
Af
Typical
Av
0.50
0.25
Figure 5.3
versus lift.
0.1
Valve flow coefficients
0.2
0.3
0.4
l/d
Lift setting screw
Fan
Cylinder
head
Laminar flow
meter
Thermocouple
Cylindrical
extension
To atmosphere
Manometer for
flow meter
Manometer for
valve pressure drop
Figure 5.4 Schematic of a steady flow bench.
0.6
Flow coefficient (Cf)
0.5
0.4
0.3
0.2
Exhaust
Intake
0.1
Figure 5.5 Intake and exhaust port
flow coefficients. Adapted from
Boretti et al. (1994).
0
0
0.1
0.2
0.3
l/d
0.4
0.5
0.6
Intake and Exhaust Flow
0.8
Discharge coefficient (Cd)
136
l/d
0.09
0.6
0.20
0.25
0.4
4
6
8
10
20
Reynolds number
40 60
104
Figure 5.6 Effect of Reynolds number and nondimensional valve lift 𝑙∕𝑑 on inlet valve discharge
coefficient (Annand and Roe, 1974).
with lift, and the representative valve area 𝜋𝑑 2 ∕4 remains constant. The maximum value
of 𝐶f is seen to be about 0.6.
The discharge coefficient 𝐶d is plotted versus Reynolds number in Figure 5.6. The
discharge coefficient 𝐶d is not a strong function of lift, since the curtain area is used to nondimensionalize the valve area in forming 𝐶d . The dependence of the discharge coefficient
𝐶d on Reynolds number in Figure 5.6 can be understood in terms of the flow patterns
shown in Figure 5.7.
At low lifts, 𝑙∕𝑑 = 0.0, the inlet jet is attached to both the valve and the seat, and
thus affected by viscous shear. The discharge coefficient, 𝐶d , decreases slightly with lift,
since the jet fills less of the reference curtain area as it transforms from an attached jet
to a separated free jet. At high lifts, 𝑙∕𝑑 ≥ 0.20, the fluid inertia prevents the flow from
turning along the valve seat, so the flow breaks away, forming a free jet. The flow area of
a free jet is more or less independent of viscosity, thus the flow coefficient at high lifts is
independent of Reynolds number.
Discharge coefficient results for exhaust valves are shown in Figure 5.8. Exhaust flow
patterns are presented in Figure 5.9, and are basically unchanged as the exhaust valve
opens, so the discharge coefficient is a weak function of the exhaust valve lift. Separation
High lift free
jet formed
(a)
Intermediate lift
Low lift jet fills gap
(b)
(c)
Figure 5.7 Flow patterns through an inlet valve (Annand and Roe, 1974).
Valve Flow
137
Discharge coefficient (Cd)
1.0
Pup /Pdown
0.8
1.01
0.6
1.65
0.4
0.2
0
0
0.1
0.2
l/d
0.4
0.3
Figure 5.8 Effect of valve lift on exhaust valve discharge coefficient (Annand and Roe, 1974).
Figure 5.9 Flow patterns
through an exhaust valve
(Annand and Roe, 1974).
(a) Low lift
(b) High lift
of the exhaust jet from the valve seat at high lift will cause the discharge coefficient to
decrease slightly at high lifts.
EXAMPLE 5.1
Exhaust Mass Flow Rate
What is the initial mass flow rate through an exhaust valve, if the valve curtain area 𝐴v
is 2.7 × 10−3 m2 , the valve discharge coefficient 𝐶d is 0.6, and the cylinder pressure and
temperature are initially at 500 kPa and 1000 K? Assume the exhaust system pressure is
105 kPa, 𝛾 = 1.35, and 𝑅 = 287 J/(kg K).
SOLUTION First compute the pressure ratio and compare it to the critical pressure ratio to determine if the flow is choked:
𝑃o
500
=
= 4.76
𝑃exh
105
(
𝑃o
𝑃exh
)
cr
Therefore, the flow is choked.
=
(
2
𝛾 +1
)−𝛾∕(𝛾−1)
= 1.86
138
Intake and Exhaust Flow
Second, use the choked flow equation, Equation 5.12, to compute the initial mass flow
rate:
)(𝛾+1)∕2(𝛾−1)
(5.17)
500 × 103
= 1.74 kg∕m3
287 × 1000
(5.18)
𝑚̇ = 𝑚̇ cr = 𝜌o 𝐶d 𝐴v 𝑐o
𝜌o = 𝑃o ∕𝑅𝑇o =
(
2
𝛾 +1
𝑚̇ = (1.74)(0.6)(2.7 × 10−3 )(1.35 × 287 × 1000)1∕2 (2∕2.35)3.36 = 1.02 kg∕s
(5.19)
Exhaust Gas Blowdown
In Chapter 2, we approximated the exhaust gas blowdown through a valve or port as a
constant volume isentropic process. Using the unsteady mass and energy conservation
equations, we can estimate a blowdown time, and compare to typical engine timescales.
If we define the control volume as the cylinder volume with instantaneous mass 𝑚(𝑡),
pressure 𝑃 (𝑡), temperature 𝑇 (𝑡), and exiting (𝑒) mass flow through an effective area 𝐴f then
𝑑𝑚
(5.20)
| = −𝑚̇ 𝑒
𝑑𝑡 𝑐𝑣
and if we assume no work or heat transfer during this process, the energy equation is
𝑑𝐸
(5.21)
| = −𝑚̇ 𝑒 ℎ𝑒
𝑑𝑡 𝑐𝑣
Assuming an ideal gas with constant specific heats, at any time the energy in the control
volume is
𝐸|𝑐𝑣 = 𝑚𝑐𝑣 𝑇
(5.22)
and the enthalpy of the exiting flow is
ℎ𝑒 = 𝑐𝑝 𝑇
(5.23)
𝑑𝑚
𝑑𝑇
𝑑
(𝑚𝑐𝑣 𝑇 ) = 𝑐𝑣 𝑇
+ 𝑐𝑣 𝑚
= −𝑚̇ 𝑒 𝑐𝑝 𝑇
𝑑𝑡
𝑑𝑡
𝑑𝑡
(5.24)
The energy equation is then
or
𝑑𝑇
(5.25)
= −𝑚̇ 𝑒 (𝑐𝑝 − 𝑐𝑣 )𝑇 = −𝑚̇ 𝑒 𝑅𝑇
𝑑𝑡
If we assume the pressure differences are great enough across the valve to produce choked
flow, then the exiting flow rate 𝑚̇ 𝑒 is
𝑚𝑐𝑣
𝑃
𝑚̇ 𝑐𝑟 = 𝐾𝐴f √
𝑅𝑇
(5.26)
Upon substitution into the energy equation,
√
𝑑𝑇
= −𝐾𝐴f 𝑃 𝑅𝑇
𝑑𝑡
Since 𝑃 = 𝜌𝑅𝑇 , 𝑅∕𝑐𝑣 = 𝛾 − 1, and 𝑚 = 𝜌𝑉 , Equation 5.27 becomes
[
]
𝐾(𝛾 − 1)𝑅1∕2 𝐴f
𝑑𝑇
=−
𝑇 3∕2
𝑑𝑡
𝑉
𝑚𝑐𝑣
(5.27)
(5.28)
Valve Flow
139
If we let the initial temperature in the cylinder be 𝑇 = 𝑇i at 𝑡 = 0, and integrate to time
𝑡, with 𝑉 = constant, then
)−2
𝑇 (𝑡) (
= 1 + 𝐶 𝑇i 𝑡
(5.29)
𝑇i
where
𝐾(𝛾 − 1)𝑅1∕2 𝐴f
2𝑉
Assuming an isentropic blowdown process,
( )𝛾∕(𝛾−1)
𝑃
𝑇
=
𝑃i
𝑇i
𝐶=
(
√ )−2𝛾∕(𝛾−1)
= 1 + 𝐶 𝑇i 𝑡
solving for time t,
𝑡=
EXAMPLE 5.2
(𝑃 ∕𝑃i )(1−𝛾)∕2𝛾 − 1
√
𝐶 𝑇i
(5.30)
(5.31)
(5.32)
Characteristic Exhaust Blowdown Time
An engine operates at 2000 rpm. The cylinder volume 𝑉 is 7.85 × 10−4 m3 when the exhaust
valve opens to ambient conditions. The average valve effective area 𝐴f is 2.50 × 10−3 m2 .
The cylinder pressure 𝑃 and temperature 𝑇 when the exhaust valve opens are at 500 kPa
and 1500 K. Estimate the time 𝑡 required for the cylinder pressure to be reduced to 200 kPa,
and compare it to the time required for one stroke. Assume 𝛾 = 1.35, and 𝑅 = 287 J/(kg K).
SOLUTION Assuming a constant cylinder volume and choked flow conditions, so Equation 5.32
is applicable. We first compute the constants 𝐾 and 𝐶 in Equation 5.32.
)(𝛾+1)∕2(𝛾−1)
(
2
= 0.676
(5.33)
𝐾 = 𝛾 1∕2
𝛾 +1
𝐶=
𝐾(𝛾 − 1)𝑅1∕2 𝐴f
2𝑉
√
(0.676)(1.35 − 1)( 287)(2.5 × 10−3 )
=
2(7.85 × 10−4 )
(5.34)
= 6.38
𝑡=
=
(𝑃 ∕𝑃i )(1−𝛾)∕2𝛾 − 1
√
𝐶 𝑇i
(200∕500)−0.13 − 1
√
6.38 1500
= 0.51 × 10−3
≈ 1∕2 ms
(5.35)
140
Intake and Exhaust Flow
A piston stroke is 180◦ , and an engine speed of 2000 rpm is equivalent to 12,000◦ /s,
so the time required for one stroke is 180/12,000 or 15 ms. For this engine, the blowdown
from 500 to 200 kPa is thus about 1/30 of the piston stroke time, justifying the constant
volume assumption.
Valve Mach Index
In this section, we examine the effect of valves on the engine volumetric efficiency. If we
assume that the pressure in the intake and exhaust ports is equal to the pressure in the intake
and exhaust plenums, respectively, then the mass inducted during the valve open period is
[
(( )
( )(𝛾+1)∕𝛾 )]1∕2
𝜃ic
𝜃ic
𝑃v
𝑃v 2∕𝛾
1
2
1
𝑑𝜃 (5.36)
𝑚𝑑𝜃
̇
=
𝜌𝐴f 𝑐
−
𝑚i =
𝜔 ∫𝜃io
𝜔 ∫𝜃io
𝛾 −1
𝑃o
𝑃o
where 𝜃io is the crank angle at which the intake valve opens and 𝜃ic is the angle at which
it closes. The terms 𝐴f , 𝜌, 𝑐, 𝑃v ∕𝑃o , and 𝛾 depend on the direction of flow, whether into or
out of the cylinder. Let us normalize Equation 5.36 by the average effective intake flow
area, 𝐴̄f ,
𝐴̄ f =
𝜃
ic
1
𝐴 𝑑𝜃 = 𝐶̄f 𝐴v
𝜃ic − 𝜃io ∫𝜃io f
(5.37)
the intake plenum density, 𝜌i , and sound speed, 𝑐i :
The term 𝐶̄f is the average flow coefficient. We then have for the volumetric efficiency
[
(( )
( )(𝛾+1)∕𝛾 )]1∕2
𝑃v
𝑃v 2∕𝛾
𝑚i
𝐴̄ f 𝑐i 𝜃ic 𝜌 𝐴f 𝑐
2
=
𝑑𝜃 (5.38)
−
𝑒v =
𝜌i 𝑉 d
𝜔𝑉d ∫𝜃io 𝜌i 𝐴̄ f 𝑐i 𝛾 − 1
𝑃o
𝑃o
Note that in the absence of reverse flow during induction 𝜌∕𝜌i = 𝑐∕𝑐i = 1; the terms
are included as shown to cover the more general case in which reverse flow occurs. A more
rigorous analysis would include the effect of engine speed.
Let us consider the limiting case in which the flow is always choked and into the
cylinder. The pressure ratio given by Equation 5.11 is independent of crank angle, so
(
)(𝛾+1)∕2(𝛾−1) ̄
𝐴f 𝑐i
2
(𝜃 − 𝜃io )
(5.39)
𝑒v =
𝛾 +1
𝜔𝑉d ic
We define a cycle averaged Mach number as the Mach index 𝑍 (Taylor, 1985) as
𝑍=
𝜋 2
𝑏
4
𝑈̄ p
𝐴̄ f 𝑐i
so the volumetric efficiency becomes, assuming 𝛾 = 1.4,
(
)
𝜃ic − 𝜃io 1
𝑒v = 0.58
𝜋
𝑍
(5.40)
(5.41)
Experimental results are available for an engine in which (𝜃ic − 𝜃io )∕𝜋 = 1.3 and thus
𝑒v = 0.75∕𝑍. They are given in Figure 5.10. Equation 5.39 is an upper bound for the
volumetric efficiency valid for large Z. It should be noted that the Mach index is not a
parameter that characterizes an actual gas speed; rather, it characterizes what the average
gas speed through the inlet valve would have to be to realize complete filling of the cylinder
Valve Flow
141
0.9
0.8
0.75/ Z
ev
0.7
0.6
0.5
0.4
0.5
1.0
1.5
Z
Figure 5.10 Volumetric efficiency versus inlet valve Mach index in the regime where choking
occurs at the inlet valve (Livengood and Stanitz, 1943).
at that particular piston speed. The Mach number for that average inlet gas speed would be
𝑍∕0.58 for 𝛾 = 1.4.
The results in Figure 5.10 show that for good volumetric efficiency one should keep
the Mach index down to less than about 𝑍 = 0.6. Based on the analyses that led to
Equation 5.41, we can interpret this to mean that the average gas speed through the inlet
valve should be less than the sonic velocity, so that the intake flow is not choked. Hence,
inlet valves can be sized on the basis of the maximum piston speed for which the engine is
designed. If we choose 𝑍 = 0.6 at this speed, it follows that the average effective area 𝐴̄ i
of the intake (𝐴̄ f → 𝐴̄ i ) valves is
𝐴̄ i ≥ 1.3 𝑏2
𝑈̄ p
𝑐i
intake
(5.42)
Likewise, for efficient expulsion of the exhaust gas, the average effective area 𝐴e , of
the exhaust (𝐴̄ f → 𝐴̄ e ) valves should be such that their Mach index is less than about 0.6,
in which case, relative to intake conditions is
( )1∕2
𝐴̄ e
𝑐i
𝑇i
≈
=
(5.43)
̄
𝑐e
𝑇e
𝐴i
As suggested by the 𝑐i ∕𝑐e ratio in Equation 5.43, a smaller exhaust valve diameter and
lift (𝑙 ∼ 𝑑∕4) can be used because the speed of sound is higher in the exhaust gases than in
the inlet gas flow. Current practice dictates that the exhaust to intake valve area ratio is on
the order of 70--80%.
In many situations, it turns out that the intake valves are sized as large as possible while
being consistent with Equation 5.43. This is because there is only so much room available
for valves and it may not be possible to satisfy Equation 5.42, thereby compromising the
maximum speed of the engine.
142
Intake and Exhaust Flow
0.12b
0.12b
0.12b
0.03b
(all)
0.38b
Ex
0.44b
In
0.49b
In
0.12b
(a)
0.30b
Ex
0.33b
In
0.133b
0.12b
0.30b
Ex
0.33b
In
0.29b
Ex
0.133b
0.29b
Ex
0.12b
(b)
(c)
Figure 5.11 Valve diameter ratios for a flat cylinder head (b:bore, In: intake, Ex: exhaust).
Adapted from Taylor (1985).
The use of multiple valves increases the valve area per unit piston area, and hence the
speed at which the engine power becomes flow limited. Heads are often wedge-shaped or
domed to increase the valve area to piston area, so that intake valve area to piston area
ratios of up to 0.5 can be obtained. Typical valve diameter ratios for two, three, and four
valves per cylinder are given for a flat cylinder head in Figure 5.11.
EXAMPLE 5.3
Intake Valve Area
What is the intake valve area 𝐴v and the ratio of the intake valve area to piston area required
to have a Mach Index 𝑍 = 0.6 for an engine with a maximum speed of 8000 rpm, bore and
stroke of 0.1 m, and inlet air temperature of 330 K? Assume 𝛾 = 1.4, 𝑅 = 287 J∕kgK, and
an average flow coefficient 𝐶̄f = 0.35.
SOLUTION
𝑐i =
√
√
𝛾𝑅𝑇o = (1.4)(287)(330) = 364 m∕s
𝑈̄ p = 2𝑁𝑠 = 2(0.1)(8000∕60) = 26 m∕s
𝐴̄ i = 1.3 𝑏2
𝑈̄ p
𝑐i
= (1.3)(0.1)2 26∕364 = 9.3 × 10−4 m2
Therefore,
𝐴v = 𝐴̄ i ∕𝐶̄f = 2.65 × 10−3 m2
𝐴v
2.65 × 10−3
= 0.34
= 𝜋
𝐴p
(0.1)2
4
In a four-stroke engine, the pumping work is defined as the work required to push
out the exhaust gas and to pull in the fresh charge. It is evaluated from a pressure--volume
diagram from bottom center at the start of the exhaust stroke to bottom center at the end
of the induction stroke. The ideal model, valid for engines operated at low Mach indices,
predicts that the pmep is the difference between the exhaust pressure and the inlet pressure.
At higher speeds, the pressure difference across the valves at closing reduces or increases
the pumping depending on whether or not the engine is supercharged. Data given by
Valve Flow
143
1.4
Valve displacement (cm)
1.2
intake
valve
1
0.8
exhaust
valve
0.6
0.4
overlap
period
0.2
Figure 5.12 Representative
exhaust and intake valve
profiles.
0
0
100
200
300
400
500
600
700
Crank angle (°)
Taylor (1985) for an engine with short intake and exhaust pipes are correlated by the
following expression:
pmep = (𝑃e − 𝑃i ) − (1.4 𝑃e − 2.6 𝑃i ) 𝑍 1.5
(5.44)
As 𝑍 → 0, the pmep goes to the ideal case of 𝑃e − 𝑃i .
Valve Timing
Intake and exhaust valve lifts are plotted as a function of crank angle in Figure 5.12. In
order to ensure that a valve is fully open during a stroke for good volumetric efficiency,
the valves are open for longer than 180◦ . The exhaust valve will open before bottom dead
center and close after top dead center. Likewise, the intake valve will open before top dead
center and close after bottom dead center. There is a valve overlap period at top dead center
where the exhaust and intake valves are both open. This creates a number of flow effects.
With a spark ignition engine at part throttle, there will be back flow of the exhaust into the
inlet manifold since the exhaust pressure is greater than the throttled intake pressure. This
will reduce the part load performance since the volume available to the intake charge is less,
reducing the volumetric efficiency. Rough idle can also result due to unstable combustion.
On the other hand, since this dilution will reduce the peak combustion temperatures, the
NO𝑥 pollution levels will also be reduced.
At wide open throttle, with both valves open, there will be some short-circuiting of the
inlet charge directly to the exhaust, since in this case, the intake pressure is greater than the
exhaust pressure. This will reduce the full load performance, since a fraction of the fuel is
not burning in the cylinder.
Typical valve timing angles for a conventional and a high-performance automobile
spark ignition engine are given in Table 5.1. The high-performance engine operates at much
higher piston speeds at wide open throttle, with power and volumetric efficiency as the
important factors, whereas the conventional engine operates at lower rpm, with idle and
part load performance of importance. Therefore, the high-performance intake valve opens
about 25◦ before the conventional intake valve, and closes about 30◦ after the conventional
intake valve. As the engine design speed increases, to maintain a maximum valve opening
during the intake and exhaust strokes, the intake and exhaust valves are open for a longer
144
Intake and Exhaust Flow
Table 5.1 Representative Valve Timing Angles for Conventional and High-Performance
Automobile Engines
Open
Intake
5◦ before tdc
30◦ before tdc
45◦ before bdc
70◦ before bdc
Conventional
High performance
Conventional
High performance
Exhaust
Close
45◦
75◦
10◦
35◦
Duration
after bdc
after bdc
after tdc
after tdc
230◦
285◦
235◦
285◦
duration, from about 230◦ to about 285◦ . Early opening of the exhaust valve will reduce
the expansion ratio, but will also reduce the exhaust stroke pumping work.
To minimize engine size and produce a given torque versus speed curve (with torque
proportional to the volumetric efficiency at fixed thermal efficiency), it is clearly desirable
to be able to vary valve timing with engine speed. Variable valve timing (VVT) is a
technique that can address the problem of obtaining optimal engine performance over a
range of throttle and engine speed. VVT allows the intake and exhaust valves to open and
close at varying angles, depending on the speed and load conditions. At idle, with a nearly
closed throttle, the intake and exhaust valve overlap is minimized to reduce exhaust back
flow. At low speed, the intake valves are closed earlier to increase volumetric efficiency and
torque. At high speed, with an open throttle, the intake valves are closed later to increase
volumetric efficiency and power.
There have been a number of VVT mechanisms built and commercialized. As one
might expect, the mechanical components of a VVT device are complex. Hydraulic mechanisms, dual lob camshafts with followers have been developed, and electromagnetic and
electrohydraulic actuators that replace the camshaft have also been used, however, at greater
expense.
Effect of Valve Timing on Volumetric Efficiency and Residual Fraction
The first law of thermodynamics applied to an open system doing boundary work is
Δ𝐸 = −
∫
𝑃 𝑑𝑉 +
∫
(𝑚̇ in ℎin − 𝑚̇ out ℎout ) 𝑑𝑡 +
∫
̇
𝑄𝑑𝑡
(5.45)
For an ideal gas with constant specific heat, it can be shown that
Δ𝐸 = 𝑐𝑣 Δ(𝑚𝑇 ) =
1
Δ(𝑃 𝑉 )
𝛾 −1
(5.46)
Let us assume that during overlap, residual exhaust gas flows up into the intake
manifold and later an equal amount flows back into the cylinder. The flow into the cylinder
is initially composed of residual exhaust gas until all of the residual gases in the intake
manifold return back to the cylinder. It follows that for the intake process
ic [
∫io
]
(𝑚𝑐
̇ 𝑝 𝑇 )in − (𝑚𝑐
̇ 𝑝 𝑇 )out 𝑑𝑡 =
ec
∫io
is
[
]+
∫ec
[
]+
∫is
ic
[ ]
(5.47)
The integrand is the same in all integrals and is abbreviated on the right-hand side by
brackets to save space. The event notation is
io = intake valve opens
ic = intake valve closes
Valve Flow
145
ec = exhaust valve closes
is = intake of fresh mixture starts
The first integral on the right-hand side of Equation 5.47 has a positive component of
enthalpy flow into the cylinder from the exhaust port and a negative component of enthalpy
flow from the cylinder to the intake manifold. (We are neglecting the small decrease in
temperature due to heat loss while the gas is in the intake port or pipe.) The second integral
is equal to the amount of enthalpy that flows into the cylinder from the exhaust port during
the overlap period, since we assume that this gas returns prior to the start of gas induction.
Hence,
[
]
ec
ic
𝑃ic 𝑉ic − 𝑃io 𝑉io = (𝛾 − 1) −
∫io
𝑃 𝑑𝑉 +
∫io
(𝑚𝑐
̇ 𝑝 𝑇 )ov 𝑑𝑡 + 𝑐𝑝 𝑇i
∫is
ic
ic
𝑚̇ in 𝑑𝑡 +
∫io
𝑄̇ 𝑑𝑡
(5.48)
Let us introduce the mass inducted, 𝑚i
𝑚i =
∫is
ic
𝑚̇ in 𝑑𝑡
(5.49)
and the mass of exhaust, 𝑚ov that flows into the cylinder from the exhaust system during
overlap
ec
𝑚ov =
∫io
𝑚̇ ov 𝑑𝑡
(5.50)
We then have for the volumetric efficiency
𝑒v =
ic
𝑇 𝑚
𝑚i
𝛾 −1 𝑄
1 𝑃ic 𝑉ic − 𝑃io 𝑉io 𝛾 − 1
𝑃 𝑑𝑉
=
+
−
− ov ov
𝜌i 𝑉 d
𝛾
𝑃i 𝑉d
𝛾 ∫io 𝑃i 𝑉d
𝛾 𝑃i 𝑉d
𝑇i 𝜌 i 𝑉 d
(5.51)
Both heat transfer to the gas and gas exchange during the overlap period decrease the
volumetric efficiency. Now let us consider the limiting case in which the piston speed is
small, 𝑈p → 0. In this case there should be no pressure drop across the valves at closure,
therefore
𝑈p → 0 ⇒ 𝑃io = 𝑃e and 𝑃ic = 𝑃i
(5.52)
Equation 5.51 then becomes
𝑒v =
𝑉ic − 𝑉io 1
−
𝑉d
𝛾
(
)
𝑉io 𝛾 − 1 𝑄
𝑇 𝑚
𝑃e
−1
−
− ov ov
𝑃i
𝑉d
𝛾 𝑃i 𝑉d
𝑇i 𝜌 i 𝑉 d
(5.53)
For engines with a short stroke to rod ratio, the cylinder volume can be approximated
by
𝑟−1
𝑉
≈1+
(1 − cos 𝜃)
𝑉o
2
(5.54)
where 𝜃 is the crank angle measured from top dead center. Finally, we can write
]
cos 𝜃io − cos 𝜃ic 𝑃e ∕𝑃i − 1 [
𝑟−1
1+
𝑒v =
−
(1 − cos 𝜃io )
2
𝛾(𝑟 − 1)
2
−
𝑇 𝑚
𝛾 −1 𝑄
− ov ov
𝛾 𝑃i 𝑉d
𝑇i 𝜌 i 𝑉 d
(5.55)
146
Intake and Exhaust Flow
Z
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1.00
0.80
0.60
Piston speed (m/s)
ev
2.44
Symbols
4.88
6.10
10.4
Range of rpm
0.40
0.20
1000
1500
2400
800
1200
1920
400
600
960
Large
Medium
Small
1700
2550
4080
0
0
2
4
6
8
10
12
Piston speed (m/s)
Figure 5.13 Volumetric efficiency versus mean piston speed of the MIT geometrically similar
engines under similar operating conditions (Taylor, 1985).
Data are available for three geometrically similar engines in which the valve overlap is small and the cylinders are not much warmer than the inlet air; the overlap and
heat loss terms are negligible in this case. Figure 5.13 presents the volumetric efficiency
of those engines as a function of piston speed. The engines had similar operating conditions: 𝑟 = 5.74, 𝜙 = 1.1, 𝑃i = 0.95 bar, 𝑃e = 1.08 bar, 𝑇i = 339K, 𝑇c = 356K. Using the
specified valve timings, compression ratio, exhaust to intake pressure ratio, and a specific
heat ratio of 𝛾 = 1.4, Equation 5.55 predicts that 𝑒v → 0.78 as 𝑈p → 0. The prediction
is seen to be quite good and similar agreement would be realized for different values of
pressure ratio.
Our analysis shows that opening and closing valves at angles other than top and bottom
center hurts the volumetric efficiency as the piston speed 𝑈p → 0. Why then are valves
opened earlier and closed later than the ideal case? In addition to the finite valve opening
and closing times discussed above, one needs to also consider that this analysis is only
valid in the limit of zero piston speed.
For a finite piston speed there will be a pressure drop across the valves, the most
important of which is at the intake valve at closing. In the limiting case, air is pushed out
of the cylinder as the piston moves up during the time from bottom dead center to inlet
valve closure. However, at a finite engine speed, the cylinder pressure at bottom center will
be less than the inlet pressure because of the pressure drop across the valve as the charge
was entering. Hence, as the piston begins the compression stroke, mixture can continue to
flow into the cylinder until the pressure rises because of the filling and the upward moving
piston. The flow will reverse itself when the two pressures are equal, and then flow back
into the intake system until valve closure.
The volumetric efficiency increases with piston speed until a point is reached where
the flow reversal starts at intake valve closure. For speeds beyond that point, volumetric
Intake and Exhaust Flow
147
efficiency will drop because the valve will close during a time in which mixture is still
flowing in the engine. The trend of volumetric efficiency with speed discussed here is
shown very clearly in Figure 5.13.
Now consider the exhaust process. At any instant, the energy equation can be
written as
(
)
𝑑𝑇
𝑑𝑉
𝑑𝑚
𝑑𝑚
(5.56)
= 𝑐𝑣 𝑚
+
𝑇 +
𝑐 𝑇
−𝑄̇ 𝑙 − 𝑃
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡 𝑝
Combined with the equation of state and integrated, we have
(
)
(
)
io 𝑄̇
𝑚io
𝑃i 1∕𝛾 𝑉io
𝑙
=
exp
𝑑𝑡
(5.57)
𝑓=
∫eo 𝑃 𝑉
𝑚eo
𝑃eo
𝑉eo
Integration is carried only to the time when the intake valve opens because during
overlap it is assumed that the gases are pushed into the intake manifold only to return
later. Notice that heat loss increases the residual fraction and is important here because the
exhaust gases are considerably hotter than the cylinder walls.
5.3 INTAKE AND EXHAUST FLOW
In engines, the configuration of the inlet and exhaust flow networks employed plays
an important role in determining the volumetric efficiency and residual fraction. Intake
manifolds (see Figure 5.14) consisting of plenums and pipes are usually required to deliver
the inlet air charge from some preparation device such as an air cleaner or compressor,
and exhaust manifolds are used to duct the exhaust gases to a point of expulsion, often far
removed from the engine. In multicylinder engines, manifolds are used so that cylinders
can share the same compressor, muffler, and catalytic converter. The flow in the inlet and
exhaust manifolds is unsteady due to the periodic piston and valve motion. The opening and
closing of the intake and exhaust valves or ports create finite amplitude compression and
rarefaction pressure waves that propagate at sonic velocity through the intake and exhaust
airflow. Computed pressure and frequency profiles in an intake manifold are shown in
Figure 5.15 in an engine operating at a low speed (≈ 3000 rpm), and in Figure 5.16 at a
higher speed (≈ 6000 rpm). As the engine speed increases, the frequency and amplitude of
the pressure waves increase proportionally.
Inlet and exhaust manifolds are sized or tuned to use the pressure waves to optimize the
volumetric efficiency at a chosen engine speed. A tuned intake manifold will have a locally
higher pressure when the intake valve is open, increasing the charge density. Likewise, a
Figure 5.14 Automotive engine
intake manifold. (Courtesy
Brodix, Inc.)
Intake and Exhaust Flow
Pressure (bar)
1.2
1.1
1.0
0.9
0.8
0
10
20
30
40
50
Time (ms)
Figure 5.15 Intake manifold
pressure and frequency at low
speed (≈ 3000 rpm). Adapted
from Silvestri et al. (1994).
Norm. amplitude
1.00
1.75
1.50
0.25
0.00
0
200
400
600
Frequency (Hz)
800
1000
0
10
20
40
50
800
1000
Pressure (bar)
1.2
1.1
1.0
0.9
0.8
30
Time (ms)
1.00
Figure 5.16 Intake manifold
pressure and frequency at high
speed (≈ 6000 rpm). Adapted
from Silvestri et al. (1994).
Norm. amplitude
148
1.75
1.50
0.25
0.00
0
200
400
600
Frequency (Hz)
tuned exhaust manifold will have a locally lower pressure when the exhaust valve is open,
increasing the exhaust outflow.
Acoustic analytical models of inlet and exhaust flow have been developed. The acoustic
analyses assume that valve opening and closing produces infinitesimal pressure waves
traveling at the speed of sound 𝑐o . A representative acoustic equation relating engine rpm,
𝑁t , to a tuned intake runner length, 𝐿t , is given by Equation 5.58:
𝑁t = 𝑎 𝑐o ∕𝐿t
(5.58)
Intake and Exhaust Flow
149
where 𝐿t is the tuned intake runner length, and 𝑎 ≃ 7.5.
The inlet airflow in a single-cylinder four-stroke engine can also be modeled as a
Helmholtz resonator with an effective volume of
𝑉 𝑟+1
(5.59)
𝑉ef f = d
2 𝑟−1
The resonant tuning rpm 𝑁t of the inlet pipe of length 𝐿i and diameter 𝐷i of a
Helmholtz resonator is given by Equation 5.60:
1∕2
2⎞
⎛1
15 ⎜ 4 𝜋𝐷i ⎟
𝑐o
𝑁t =
𝜋 ⎜ 𝐿i 𝑉ef f ⎟
⎝
⎠
(5.60)
where 𝐿i is an effective length from the inlet valve to the atmosphere and 𝐷i is an effective
diameter that with 𝐿i matches the inlet system volume.
The sensitivity of the volumetric efficiency to runner length and engine speed has been
a challenge to engine design engineers. By choosing an intake runner of a given length,
the volumetric efficiency can be increased for a particular engine speed, but it drops off
more sharply at other speeds. A fixed length intake runner is desirable in engines with
throttle body injectors or carburetors to minimize wall wetting and maldistribution of the
fuel air mixture. With port fuel injection it is possible to use a variable runner length,
and production engines with variable intake runner length are now becoming common in
vehicles.
A representative plot of the effect of inlet runner length on the volumetric efficiency
of an automobile engine is plotted as a function of engine speed in Figure 5.17. As the
engine speed decreases, the volumetric efficiency approaches 0.8, and with no intake pipe
or runner, the volumetric efficiency follows a shallow curve with a maximum at about 0.9,
consistent with the analyses presented earlier in this chapter, in which the runner length
for maximum volumetric efficiency is predicted to be inversely proportional to the engine
speed.
Gas dynamics codes are used for the design of intake and exhaust plenums, runners,
and ports, and to assess the effect of design changes on the flow patterns and the volumetric
efficiency. There are a number of 1-D gas dynamics programs, such as WAVE (WAVE
USER’S MANUAL, 2014) and GT-POWER (GT-POWER USER’S MANUAL, 2014),
that numerically solve the compressible governing equations to predict intake and exhaust
system pressure and flow profiles over an engine cycle. The governing equations for
Lp (mm)
780
680
520
No pipe
Figure 5.17 Volumetric
efficiency versus engine
speed and intake runner
length. Adapted from
Tabaczynski (1982).
150
Intake and Exhaust Flow
Figure 5.18 CFD grid for intake manifold
flow. (Courtesy Adapco.)
the intake and exhaust flow are the unsteady mass, momentum, and energy conservation
equations, which for one dimension in vector form are
𝜕 𝑓⃗ 𝜕 𝐹⃗
+
= 𝑇⃗
𝜕𝑡
𝜕𝑡
⎡ 𝜌𝐴 ⎤
⎥
⎢
⃗
𝑓 = ⎢ 𝜌𝐴𝑈 ⎥
⎢ 𝜌𝐴𝐸 ⎥
⎦
⎣
⎡
⎤
𝜌𝐴𝑈
⎢
⎥
2
𝐹⃗ = ⎢ 𝜌𝐴(𝜌𝑈 + 𝑃 ) ⎥
⎢ 𝜌𝐴𝑈 𝐻 ⎥
⎣
⎦
0
⎤
⎡
√
⎢
𝛿𝐴 ⎥
⃗
𝑇 = ⎢ 𝜏w 4𝜋𝐴 + 𝑃 𝛿𝑥 ⎥
√
⎥
⎢
⎦
⎣
𝑞w 4𝜋𝐴
(5.61)
(5.62)
(5.63)
(5.64)
where 𝜌 is the fluid density, 𝐴 is the cross-sectional area of the duct, 𝑈 is the fluid velocity,
𝑃 is the pressure, 𝐸 is the total specific energy, 𝐻 is the total specific enthalpy, 𝜏w is the
wall shear stress, and 𝑞w is the wall heat flux. Physical effects such as wall friction, heat
transfer, area changes, branches, and bends that occur in actual manifolds can be accounted
for in the above equations.
The specific boundary condition geometry of the intake and exhaust ports is required
for solution of Equations 5.61--5.64. A representative three-dimensional intake manifold
geometry divided into computational mesh elements is shown in Figure 5.18. The modeling
includes the intake manifold, the intake ports, valves, and cylinder volumes. The computed
velocities with the middle cylinder open are shown in Figure 5.19. The CFD grid for the
exhaust manifold shown in Figure 5.20 is for a three-cylinder engine with a closely coupled
catalytic converter. The computed velocities with the middle cylinder exhaust valve open
are shown in Figure 5.21.
5.4 SUPERCHARGERS AND TURBOCHARGERS
The power and efficiency of an internal combustion engine can be increased with the
use of an air compression device such as a supercharger or turbocharger. Increasing the
pressure and density of the inlet air will allow additional fuel to be injected into the cylinder,
increasing the power produced by the engine. Spark ignition engines are knock limited,
Superchargers and Turbochargers
151
V6 Manifold (samm Mesh)
STAR
PROSTAR 3.10
VELOCITY MAGNITUDE
M/S
ITER = 688
LOCAL MX = 51.59
LOCAL MN = 0.7094-E-04
Z
Figure 5.19 CFD velocity
results for intake manifold
flow. (Courtesy Adapco.)
X
Y
51.59
47.91
44.22
40.54
36.85
33.17
29.48
25.80
22.11
18.43
14.74
11.06
7.370
3.685
0.7248E-04
Figure 5.20 CFD grid for exhaust
manifold flow. (Courtesy Adapco.)
restricting the allowable compressor pressure increase. An intercooler heat exchanger is
used with turbochargers and superchargers to cool the intake air and increase its density
after the compression process has raised its temperature, and reduce the tendency to knock.
Superchargers and turbochargers are used extensively on a wide range of diesel engines,
since they are not knock limited.
As shown in Figure 5.22, superchargers are classified as compressors that are mechanically driven off of the engine crankshaft. Superchargers are used in applications in
which the increased density and pressure is desirable at all engine speeds. P. H. Roots, an
American engineer, invented the supercharger in 1859, for use in the then-emerging steel
industry. Superchargers have also been used in piston-driven airplane engines since about
1910 to compensate for the decrease in air pressure and density with altitude, and to increase
the flight ceiling. Since it is mechanically driven, the rotational speed of a supercharger is
limited to about speeds of the order of 10,000 rpm.
Turbomachinery considerations relative to engines include
• coupling compressors and turbines and matching them to the mass flow rate of the
internal combustion engine;
• aftercooling of the compressed charge;
• relating steady-flow bench tests to actual periodic flow conditions; and
• transient response of the entire engine system.
152
Intake and Exhaust Flow
RIGHT-HAND EXHAUST MANIFOLD AND CATAYLIST
Case 2: Runner 2 open, Velocity = 4 m/s, other runners closed.
STAR
4.000
3.778
3.556
3.333
3.111
2.889
2.667
2.444
2.222
2.000
PROSTAR 2.1
19 Jul 92
VELOCITY MAGNITUDE
M/SEC
LOCASL MX = 5.044
LOCAL MN = .2621E-05
1.778
1.556
1.333
1.111
.8889
Z
.6667
.4444
Figure 5.21 CFD results
for exhaust manifold flow.
(Courtesy Adapco.)
.2222
Y
–.2980E-07
(a) Reciprocating
Figure 5.22 Types of positive
displacement compressors.
X
(b) Roots
(d) Lysholm screw
(c) Sliding vane
Superchargers and Turbochargers
153
4
1
Pi
Pe
Figure 5.23 Comparison of turbine
and compressor work.
b
6
a
5a
5
a: blowdown work to turbine
b: compression work
The types of compressors used on internal combustion engines are primarily of two
types: positive displacement and dynamic. With a positive displacement compressor, a
volume of gas is trapped, and compressed by movement of a compressor boundary element. Three types of positive displacement compressors are the Roots, vane, and screw
compressor, as shown in Figure 5.22. The efficiency of positive displacement compressors
varies from about 50% for the Roots compressor to over 90% for the screw compressor. A
dynamic compressor has a rotating element that adds tangential velocity to the flow which
is converted to pressure in a diffuser. Two types of dynamic compressors and turbines are
radial (centrifugal) and axial.
Turbochargers are defined as devices that couple a compressor with a turbine driven
by the exhaust gases, so that the pressure increase is proportional to the engine speed. The
turbocharger was first invented in 1906, and the applications have expanded from marine
diesel engines, to vehicle diesel engines, and then to spark ignition engines. The potential
increase in overall system efficiency with a turbocharger can be seen by inspection of
Figure 5.23, in which a portion of the available work obtained from the blowdown of the
exhaust gas can be used to compress the intake gas.
A turbocharger is a coupled dynamic compressor and dynamic turbine, due to the high
rotational speeds, of the order of 100,000 rpm, required for efficient operation at typical
internal combustion engine flow rates and pressure ratios. For automotive applications
an outward radial flow geometry is used for the compressor, and an inward radial flow
is used for the turbine. A cross section of a turbocharger with a radial compressor and
turbine is shown in Figures 5.24 and 5.25. Turbochargers are controlled by varying the
pressure ratio.
A waste gate is used to control the exhaust gas flow rate to the turbine. The waste
gate is a butterfly or poppet valve controlled by the intake manifold pressure to prevent
the turbocharger from compressing the intake air above a set knock or engine stress
pressure limit. More recently, variable nozzles have been used to adjust the nozzle area
to provide acceptable vehicular acceleration performance as well as efficient operation at
high loads.
A turbine can also be mechanically connected to the engine drive shaft, a configuration
called compounding. Turbochargers used in diesel locomotives use a clutch geared to the
drive shaft to drive the compressor during low engine speed when there is insufficient power
from the turbine. At higher engine speeds, the clutch disengages, and the compressor is
driven by the exhaust gases flowing through the turbine. Axial flow compressors and
turbines are typically used in marine turbocharger applications.
154
Intake and Exhaust Flow
Figure 5.24 Turbocharger cutaway.
(Courtesy PriceWeber.)
Intake
flow
Centrifugal
compressor
Turbine
Exhaust
flow
Figure 5.25 Turbocharger
cross section (Laustela et al.,
1995).
Diffuser
The adiabatic efficiency 𝜂c of a compressor is defined as the isentropic work required
to compress the gas over the specified pressure ratio divided by the actual work required
to compress the gas over the same pressure ratio. The pressure ratio 𝑃2 /𝑃1 of compressors
used for internal combustion engines is generally small enough that the gas may be assumed
to have constant specific heat. It follows then that the isentropic work required per unit
mass of gas to compress the gas from 𝑃1 to 𝑃2 is given by
[
]
𝑤1−2s = 𝑐𝑝 (𝑇1 − 𝑇2s ) = −𝑐𝑝 𝑇1 (𝑃2 ∕𝑃1 )(𝛾−1)∕𝛾 − 1
(5.65)
In deriving Equation 5.65, it was tacitly assumed that the change in kinetic energy across
the compressor was negligible compared to the change in enthalpy, an assumption usually
valid in practice.
Experiments with compressors show that the adiabatic efficiency is dependent primarily upon the pressure ratio, the tip Mach number, and the mass flow rate ratio,
Superchargers and Turbochargers
155
given below:
𝑃2 ∕𝑃1
𝑠∕𝑐o
𝑚∕
̇ 𝑚̇ cr
Outlet-inlet pressure ratio
Mach number based on rotor tip speed, 𝑠 = 𝜔𝐷∕2
Ratio of the mass flow rate to the critical mass flow rate
As shown previously in this chapter, the critical mass flow rate for choking flow with
𝛾 = 1.4 is
(
)
𝜋𝐷2
𝑚̇ cr = 0.578
(5.66)
𝜌o 𝑐o
4
Compressor performance characteristics are plotted on a compressor map, with mass
flow rate on the 𝑥 axis and pressure rise on the 𝑦 axis. Lines of constant adiabatic efficiency
and constant rotational speed are plotted as a function of mass flow rate and pressure ratio.
Note that the constant speed lines have a negative slope on the mass flow rate--pressure
plane. Performance data for various piston, Roots, Lysholm, screw, centrifugal, and axial
compressors are given in Figures 5.26--5.29. It can be seen that the various compressor configurations occupy different regions of the compressor map, allowing different compressors
to be matched to a given conditions.
Dynamic compressors have surge and choking performance limits. The surge limit on
the left side of the dynamic compressor map represents a boundary between stable and
unstable operating points. For stable operation dynamic compressors operate to the right of
the surge line, with a negative slope to the constant speed lines. Surge is a self-sustaining
flow oscillation. When the mass flow rate is reduced at constant pressure ratio, a point
arises where somewhere within the internal boundary layers on the compressor blades a
flow reversal occurs. If the flow rate is further reduced, then a complete reversal occurs
that relieves the adverse pressure gradient. That relief means a flow reversal is no longer
needed and the flow then begins to return to its initial condition. When the initial condition
is reached, the process will repeat itself, creating surge.
On the right side of the dynamic compressor map is a zone where efficiencies fall
rapidly with increasing mass flow rate. The gas speeds are quite high in this zone and
the attendant fluid friction losses are increasing with the square of the gas speed. In this
region, there is also the choke limit, which occurs at a slightly different value of 𝑚∕
̇ 𝑚̇ cr
for each tip speed. Choking occurs when at some point within the compressor the flow
reaches the speed of sound. It occurs at values of 𝑚∕
̇ 𝑚̇ cr less than 1 because 𝑚̇ cr is based on
the compressor wheel diameter 𝐷 rather than on the cross-sectional area where choking is
occurring. The value of 𝑚∕
̇ 𝑚̇ cr at choking varies with tip speed because the location within
the compressor at which choking occurs depends on the structure of the internal boundary
layers.
A procedure for matching a compressor with an internal combustion engine is listed
below. Using the performance map, the compressor mass flow rate is matched to the engine
mass flow rate, given the engine volumetric efficiency as a function of engine speed. The
procedure is as follows:
(a) Assume a pressure ratio.
(b) Read the compressor efficiency and mass flow rate.
(c) Solve for the adiabatic temperature and density after compression.
(d) Calculate the engine mass flow rate with the density found in part (c) and the known
volumetric efficiency.
156
Intake and Exhaust Flow
10
8
Piston
6
4 0.90
0.20 0.24
3
P2
P1
0.28 s/co
0.80
2
1.8
1.6
0.10 0.15
0.01
nc 0.70
Lysholm
0.015 0.02 s/co
0.05
0.80
Roots
1.4
0.75 nc
0.70
0.80
1.2
n c 0.90
0.004 0.006 0.01
0.02
0.1
0.04 0.06
. .
0.2 0.3 0.4 0.6 0.8
m/m cr
(a)
4
3
s /co = 0.9
s /co = 1.2
1.1
Single-stage centrifugal
0.8
0.84
1.0
0.82
0.9
2
1.8
1.6
0.7
0.6
0.7
0.80
0.81
P2
P1
1.4
0.80
0.8
0.75
0.5
Constant s /co
Constant nc
Surge line
s /co = 0.9
10-stage
axial flow
0.8
1.2
0.7
0.6
1.1
1.08
0.5
Single-stage axial flow
1.06
Figure 5.26 Comparative
performance of various
positive displacement and
dynamic compressors
(Taylor, 1985).
0.90
1.04
0.01
0.02 0.030.04 0.06
0.1
. .
0.89
0.2
0.3 0.4
0.6 0.8
m/m cr
(b)
(e) Iterate until the engine mass flow rate and compressor flow rate are equal.
(f) Calculate the compressor power.
There are alternative devices that compete with positive displacement and dynamic
turbomachines. Such alternatives include shock wave compressors (Weber, 1995), an example of which is a device called the Comprex in which air is compressed by means of
exhaust pressure waves and momentary direct contact between the exhaust stream and the
fresh air (Gaschler, Eib, and Rhode, 1983).
157
Superchargers and Turbochargers
3.4
Schwitzer compressor with
D = 99 mm compressor wheel
and vaneless diffuser
3.2
Surge line
3.0
Compressor total pressure ratio (P2/P1)
1.34
2.8
2.6
2.4
s/co = 1.19
2.2
0.68 = nc
0.72
1.04
2.0
0.74
1.8
0.76
0.89
1.6
0.78
0.74
1.4
0.59
1.2
Figure 5.27 Centrifugal
compressor map.
(Courtesy R. Hehman of
Schwitzer.)
1.0
0
0.1
0.2
0.3
. .
0.4
Mass flow to critical flow m/mcr
6000
8000
1.7
10,000 12,000 rpm
4000
1.6
40
45
Pressure ratio
1.5
2000
n (%)
50
1.4
55
1.3
1.2
1.1
Figure 5.28 Representative
Roots supercharger
performance. (Adapted from
Sorenson, 1984.)
1.0
0
0.05
0.10
Mass flow (kg/s)
0.15
158
Intake and Exhaust Flow
Figure 5.29 Representative
centrifugal compressor map.
(Adapted from Andersson
et al., 1984.)
5.5 EFFECT OF AMBIENT CONDITIONS ON ENGINE AND COMPRESSOR
MASS FLOW
Engines are designed to operate over large ranges of ambient temperature, pressure, and
humidity. The ambient pressure is to first order determined by the altitude above sea level.
Engine tests are corrected for the effects of ambient pressure and temperature using the
ideal gas compressible flow equation, Equation 5.10, restated below as Equation 5.67:
𝑚̇ a = 𝐴ef f
𝑃o
(𝑅𝑇o )1∕2
[
2𝛾
𝛾 −1
((
𝑃
𝑃o
)2∕𝛾
(
−
𝑃
𝑃o
)(𝛾+1)∕𝛾 )]1∕2
(5.67)
If 𝑃 ∕𝑃o is assumed to remain constant, then the mass flow rate through the engine
varies as
𝑃o
(5.68)
𝑚̇ a ∼
(𝑇o )1∕2
This parameter is also used as a correction factor on compressor maps to account for
varying ambient conditions. For a constant fuel--air ratio, the above equation implies, as
expressed below in Equation 5.69, that
( )
bmepm
𝑃m 𝑇o 1∕2
=
(5.69)
bmepo
𝑃 o 𝑇m
where the subscript 𝑚 denotes the measured conditions and the subscript o denotes standard
atmosphere conditions at sea level.
Two examples clearly illustrate the altitude effect on engine performance. Naturally
aspirated diesel locomotive engines are usually derated by 2.5% per 300 m of elevation
change, so at 10,000 ft (3.05 km) above sea level, the elevation of most mountain passes
in the Western United States, they will produce 25% less power. Most locomotive diesels
are now turbocharged to increase power and reduce the altitude effect.
Also, as an aircraft flies from sea level to an elevation of 20,000 ft (6.1 km), the
density of the standard atmosphere decreases by 50% from 1.22 to 0.66 kg/m3 . The bmep
References
159
1.0
Equation 5.10
Average of air-cooled
aircraft engines
bmep/bmep at sea level
0.8
0.6
0.4
Liquid-cooled aircraft engine
Two liquid-cooled aircraft engines
Four-stroke diesel engine
Two-stroke diesel engine
0.2
Figure 5.30 Effect of altitude
on unthrottled engine
performance at constant
fuel--air ratio and coolant
temperature.
0
0
2
4
6
Altitude (km)
8
10
performance of a number of naturally aspirated aircraft engines with altitude is shown in
Figure 5.30. As a result of the density decrease, there is a 60% decrease in bmep at 6000 m
relative to sea level. Equation 5.69 correlates the experimental data very well, as shown
in Figure 5.28. Many piston-driven aircraft engines are supercharged to allow suitable
operation at elevations above sea level.
For further information regarding intake and exhaust flow in engines and compressors,
the reader is referred to the texts by Blair (1998), Watson and Janota (1982), Wilson and
Korakianitis (2014), and Winterbone and Pearson (1999).
5.6 REFERENCES
ANDERSSON, J., A. BENGTSSON, and S. ERIKSSON (1984), ‘‘The Turbocharged and Intercooled 2.3 L
Engine for the Volvo 760,’’ SAE paper 840253.
ANNAND, W. and G. ROE (1974), Gas Flow in the Internal Combustion Engine, G. T. Foulis, Somerset,
England.
BLAIR, G. (1998), Design and Simulation of Four Stroke Engines, SAE International, Warrendale,
Pennsylvania.
BORETTI, A., M. BORGHI, and G. CANTORE (1994), ‘‘Numerical Study of Volumetric Efficiencies in a
High Speed, Four Valve, Four Cylinder Spark Ignition Engine,’’ SAE paper 942533.
GASCHLER, E., W. EIB, and W. RHODE (1983), ‘‘Comparison of the 3-Cylinder DI-Diesel with Turbocharger or Comprex-Supercharger,’’ SAE paper 830143.
GT-POWER USER’S MANUAL (2014), Gamma Technologies, Westmont, Illinois.
LAUSTELA, E., U. GRIBI, and K. MOOSER (1995), ‘‘Turbocharging the Future Gas and Diesel Engines
of the Medium Range,’’ ASME ICE Conf.; Vol 25-1, p. 1521.
LIVENGOOD, J. and J. STANITZ (1943), ‘‘The Effect of Inlet Valve Design, Size and Lift on the Air
Capacity and Output of a Four-Stroke Engine,’’ NACA Technical Note TN-915.
SILVESTRI, J., T. MOREL, and M. COSTELLO (1994), ‘‘Study of Intake System Wave Dynamics and
Acoustics by Simulation and Experiment,’’ SAE paper 940206.
SORENSON, S. (1984), ‘‘Simulation of a Positive Displacement Supercharger,’’ SAE paper 820244.
TABACZYNSKI, R. (1982), ‘‘Effects of Inlet and Exhaust System Design on Engine Performance,’’
SAE paper 821577.
TAYLOR, C. (1985), The Internal Combustion Engine in Theory and Practice. MIT Press, Cambridge,
Massachusetts.
160
Intake and Exhaust Flow
WATSON, N. and M. JANOTA (1982), Turbocharging the Internal Combustion Engine, Wiley,
New York.
WAVE USER’S MANUAL (2014), Ricardo Software, Inc., Burr Ridge, Illinois.
WEBER, H. (1995), Shock Wave Engine Design, Wiley, New York.
WILSON, D. and T. KORAKIANITIS (2014), The Design of High Efficiency Turbomachinery and Gas
Turbines, Second Edition, MIT Press, Cambridge, Massachusetts.
WINTERBONE, D. and R. PEARSON (1999), Design Techniques for Engine Manifolds, Society of Automotive Engineers, Warrendale, Pennsylvania.
5.7 HOMEWORK
5.1
If an engine has a bore of 0.1 m, stroke of 0.08 m, inlet flow effective area of 4.0 × 10−4 m2 ,
and inlet temperature of 320 K, what is the maximum speed it is intended to be operated
while maintaining good volumetric efficiency?
5.2
Explain how unburned fuel can appear in the exhaust during the intake and exhaust strokes.
5.3
Combustion gases (𝛾 = 1.3, 𝑅 = 280 J/(kg K)) exit through the exhaust port of a two-stroke
engine during blowdown. The exhaust port geometry can be modeled as a converging
nozzle with a port diameter of 2 cm. The cylinder gases are initially at 200 kPa and 393 K,
and 𝑃atm = 101 kPa. What is the initial velocity and mass flow rate of the exhaust flow?
5.4
It was explained in the chapter that because of the pressure drop across a valve, it is
advantageous to close the intake valve after bottom dead center. Use the same logic to
explain why exhaust valves are closed after top dead center and what the effect of engine
speed is on the residual fraction.
5.5
Suppose an engine were constructed with variable valve timing, thus ensuring optimum
timing at all speeds. Explain how the volumetric efficiency would depend on speed for
wide open throttle operation with short pipes and 𝑍 < 0.6.
5.6
Figure 5.2 shows an inlet valve opened to 𝑙∕𝑑i = 0.25. If the stem is chosen to be 𝑑𝑠 = 0.15𝑑i
and the throat of the port is 𝑑t = 0.85𝑑i , what would be the flow coefficient based purely
on the geometrical blockage?
5.7
Calculate the ratios of the inlet valve area to piston area for the three configurations a, b,
and c in Figure 5.11.
5.8
Compare the performance of a single-inlet valve and a double-inlet valve configuration.
The diameter of the inlet valve is 22 mm for the single-valve configuration, and 16 mm for
the double-valve configuration. If the maximum valve lift 𝑙∕𝑑 = 0.25, what is the difference
in the valve curtain and the valve seat areas for both cases? What are some advantages to
using four valves per cylinder?
5.9
If the inlet Mach index in each case in Figure 5.11 is held to 𝑍i = 0.6 and 𝑐i = 400 m/s,
𝐴̄ i = 0.35 𝑛i (𝜋∕4) 𝑑i2 where 𝑛i = number of intake valves, then what would be the mean
piston speed in each case?
5.10
A four-stroke four-cylinder square (bore = stroke) engine has a displacement volume of 5 L
and operates at 3000 rpm. The intake air temperature is 350 K, the intake manifold length
is 1.25 m long, and 𝐶̄f = 0.38. (a) For a Mach index 𝑍 = 0.6, what is the mean piston speed
and average effective intake valve flow area? (b) At what engine speed would the intake
manifold be ‘‘tuned’’ for increased intake mass flow?
Homework
161
5.11
Compare the predicted resonant tuning rpm 𝑁t (Equation 5.60) of a Helmholtz resonator
model with the simple acoustic tuning rpm of Equation 5.58 and also the experimental
results for maximum volumetric efficiency 𝑒v plotted in Figure 5.17. Assume 𝐷i is equal
to the inlet pipe diameter. Make a table of the tuning rpm versus tuning inlet pipe length
for the five cases shown in Figure 5.17. Assume 𝑏 = 83 mm, 𝑠 = 106 mm, 𝐷i = 0.05 m,
𝑟 = 9, 𝑇o = 300 K.
5.12
Three camshafts are available for an engine. The valve maximum lift and intake and exhaust
opening and closing angles in degrees relative to top dead center (tdc) and bottom dead
center (bdc) are tabulated below.
5.13
CAM
IO
(btdc)
IC
(abdc)
EO
(bbdc)
EC
(atdc)
LIFT
(mm)
Factory
A
B
30
26
22
60
66
62
60
66
62
30
26
22
9.5
11.4
10.3
Draw a sketch of the three cam timing diagrams. Discuss the effects these different cams
might have, including duration and overlap effects.
Derive an expression for the volumetric efficiency of a supercharged engine, using an
analysis similar to the derivation of Equation 5.51.
5.14
A supercharger has an isentropic efficiency of 0.75 and consumes 20 kW. If the volumetric
flow rate of standard air into the supercharger is 250 L/s, what is the air temperature,
pressure, and density exiting the supercharger? Assume standard inlet conditions.
5.15
Develop Equation 5.65 for the work done in an isentropic compression.
5.16
The airflow into a four-stroke 3.5 L engine operating at 3000 rpm with a volumetric
efficiency of 0.75 is to be supercharged to 145 kPa from ambient 𝑃o , 𝑇o conditions. An
intercooler cools the compressed air to 325 K. If the supercharger isentropic efficiency is
0.60, what is the power consumption of the supercharger?
5.17
A Roots supercharger map is given in Figure 5.28. Match (i.e., find the resultant pressure ratio) this supercharger to a 2.0 L, four-stroke engine with the following volumetric
efficiencies.
N (rpm)
𝑒v (%)
1000
2000
3000
4000
5000
6000
68
68
75
76
73
70
Find the power required to drive the supercharger at each condition as well as the outlet
temperature. Choose a compressor speed 𝑁c equal to twice the engine speed 𝑁.
162
5.18
Intake and Exhaust Flow
A naturally aspirated four-cylinder, four-stroke gasoline engine has the following specifications.
𝑉d
𝑏
𝑠
𝑟
𝑊̇ b
𝜏b
2316 cm3
96 mm
80 mm
9.5
83 kW at 𝑁 = 5400 rpm
184 N-m at 𝑁 = 2760 rpm
A turbocharged version of the engine utilizes the compressor mapped in Figure 5.29.
Estimate the brake power of the turbocharged engine at 𝑁 = 5400 rpm if the compressor
ratio is 𝑃2 ∕𝑃1 = 1.5. What is the cmep and bmep at this speed. What is the compressor
efficiency and rotational speed? What is the heat transfer to the inter-cooler? Make the
following assumptions.
• For the naturally aspirated (NA) engine
Inlet manifold conditions: 𝑇i = 310 K, 𝑃i = 1.0 bar, 𝜙 = 1.0.
Volumetric efficiency: 𝑒v = 0.84.
Mechanical efficiency: 𝜂m = bmep/(imep)net = 0.90.
• For the turbocharged (TC) engine
Aftercooled gas temperature: 𝑇i = 340 K. Volumetric efficiency: 𝑒v = 0.91.
Mechanical efficiency: 𝜂m = bmep/(imep)net = 0.88.
• For a given engine speed and displacement, the indicated power is proportional to
airflow rate:
𝑊̇ i ∝ imepnet ∝ 𝑒v 𝑃i ∕𝑇i
In practice, the compression ratio was lowered to 8.7 to avoid knock and the engine
produced 117 kW at 5280 rpm.
Chapter
6
Fuel and Airflow in the
Cylinder
6.1 INTRODUCTION
In this chapter, we examine the delivery and in-cylinder flow of the fuel and air in both
spark ignition and compression ignition engines. The basic principles of carburetors and
fuel injection systems are laid out, providing an introduction to the various means employed
to deliver fuel to the combustion chamber. In-cylinder flow is a large-scale turbulent mixing
process initially governed by the momentum of the incoming air and fuel during the intake
stroke, and then modified by the piston motion during the compression stroke.
Adequate mixing of the fuel and air is essential for a satisfactory combustion process
that will produce the engine power required with minimum emissions. The timescales are
very short, for example, in an automotive engine, the time available for mixing during
the compression stroke is on the order of tens of milliseconds. During this time between
injection and start of combustion, a liquid fuel needs to be broken up into droplets, vaporized,
and mixed with the surrounding air.
Design of an engine’s air--fuel mixing process is an engineering challenge as it is a
compromise of many conflicting demands. For direct-injection (DI) engines, large-scale
mixing and turbulence generation can be achieved by high-pressure fuel injection. In these
cases, the mixing patterns are governed by the momentum flux of the injected fuel. Some
engines rely on the angular momentum or swirl of the intake air for adequate mixing.
However, increased swirl reduces volumetric efficiency and increases convective heat loss.
Engines designed without swirl are termed quiescent and high-pressure fuel injection,
which entrains cylinder air, is instead relied upon to fully mix the fuel and air. However, a
higher required injection pressure will result in a more costly fuel injection system.
The chapter finishes with coverage of two-stroke engine configurations and airflow
scavenging models.
6.2 CARBURETION
Carburetors are used on spark ignition engines to control the fuel flow delivered to an
engine so that it is proportional to the airflow. As shown in Figures 6.1 and 6.2, carburetors
are used for both liquid and gaseous fuels. With liquid fuels, they also serve to mix the
fuel with the air by atomizing the liquid into droplets so that it will evaporate quickly.
The liquid fuel carburetor was invented and patented in 1893 by W. Maybach (1846--
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
163
164
Fuel and Airflow in the Cylinder
3
.
ma
P∞, T∞
Venturi
Fuel
nozzle
2
3
Fuel
.
mf
Throttle
Figure 6.1 Carburetor for mixing
liquid fuels with air.
Metering orifice
4
Figure 6.2 Carburetor for mixing gaseous
fuels with air. (Courtesy Impco, Inc.)
1929), a German engineer, and used for mixture preparation in vehicular engines until the
mid-1980s. Currently, due to emissions regulations, they are primarily used with small
(<25 kW) engines.
The basic principle behind a liquid fuel carburetor is shown in Figure 6.1, indicating
the inlet airflow through a venturi with a fuel nozzle at the throat and then past a throttle
valve. In contrast to fuel injectors, liquid fuel carburetors atomize the fuel by processes
relying on the air speed being greater than the fuel speed at the fuel nozzle. The pressure
difference between the carburetor inlet and the nozzle throat is used to meter the fuel to
achieve a desired air--fuel ratio. Therefore, the fuel is metered using the airflow as an
independent variable. The mass flow between locations (1) and (4) is determined by the
engine speed and throttle position. The pressure at location (2) and the fuel--air ratio, 𝑚̇ f ∕𝑚̇ a ,
are dependent variables that adjust themselves to match the mass flow 𝑚̇ 4 = 𝑚̇ f + 𝑚̇ a that
the engine is demanding. For starting purposes, a choke is added upstream of the venturi
to enrich the mixture. It is basically another throttle that acts to lower the air pressure and
thus the airflow rate in the carburetor, while keeping the fuel flow rate relatively constant.
Carburetion
165
Assuming steady ideal gas flow through the carburetor, the airflow is given by Equation
6.1:
[
𝑚̇ a = 𝜌∞ 𝐴a 𝑐∞
2
𝛾 −1
((
𝑃2
𝑃∞
(
)2∕𝛾
−
𝑃2
𝑃∞
)(𝛾+1)∕𝛾 )]1∕2
(6.1)
In Equation 6.1, 𝑐∞ is the speed of sound in the atmospheric air, and 𝐴a = 𝐶d, a 𝐴v is
the effective flow area at the venturi throat. That flow area is less than the venturi throat
cross-sectional area because of blockage by the fuel nozzle and boundary layers along the
venturi walls.
The fuel flow is computed assuming the fuel is incompressible, in which case
√
(6.2)
𝑚̇ f = 𝐴f 2𝜌f (𝑃∞ − 𝑃2 )
where the area 𝐴f = 𝐶d,f 𝐴orif ice is the effective flow area of the metering orifice. According
to Obert (1973), the discharge coefficient of metering orifices used in carburetors is typically
0.75; it accounts for boundary layers in the orifice and for the small pressure drop from the
orifice to the nozzle.
The maximum flow rate of air through a carburetor occurs when the flow chokes at
the venturi nozzle. In this case,
(
)𝛾∕(𝛾−1)
)
(
𝑃2
2
=
(6.3)
𝑃∞ cr
𝛾 +1
(
𝑚̇ a, cr = 𝜌∞ 𝐴a 𝑐∞
2
𝛾 +1
)(𝛾+1)∕2(𝛾−1)
(6.4)
Equation 6.4 is useful in sizing a carburetor venturi; the effective area 𝐴a is a function of
the maximum airflow rate.
Let us call the ratio of the airflow to the critical or choked airflow, the carburetor
demand 𝐷c . It should be clear that 0 ≤ 𝐷c ≤ 1. Assuming 𝛾 = 1.4, which when substituted
into Equations 6.1 and 6.4 and solved with Equation 6.2 for the fuel--air ratio FA yields
[
]
𝑚̇ f
1.73 𝜌f 𝐴f 2(𝑃∞ − 𝑃2 )
=
(6.5)
FA =
2
𝑚̇ a
𝐷c 𝜌∞ 𝐴a
𝜌f 𝑐∞
By definition, the carburetor demand 𝐷c is then
𝑚̇ a
𝐷c ≡
= 3.86
𝑚̇ a, cr
[(
𝑃2
𝑃∞
(
)1.43
−
𝑃2
𝑃∞
)1.71 ]1∕2
(6.6)
A graph of the fuel--air ratio as a function of carburetor demand is shown in Figure 6.3,
assuming typical values for gasoline properties and different values of the effective area
ratio 𝐴f ∕𝐴a . The curves are based on Equation 6.5 for 𝑃∞ = 0.987 bar, 𝜌f = 749 kg/m3 ,
𝑐∞ = 346 m/s, and 𝜌∞ = 1.17 kg/m3 . Note that for demands between 20 and 80%, the
fuel--air ratio is a weak function of demand and its value is dependent primarily upon
the geometric properties of the carburetor through the ratio 𝐴f ∕𝐴a . At demand less than
about 20%, the fuel--air ratio would in reality be much less than predicted by Equation 6.5
because of surface tension effects at the nozzle exit. The simple carburetor just described
can then be expected to operate only over the demand range 0.20 < 𝐷c < 0.80.
166
Fuel and Airflow in the Cylinder
0.10
Af
Aa
Fuel–air ratio
0.08
0.0030
0.0025
0.06
0.0020
0.04
0.02
Figure 6.3 The fuel--air ratio
as a function of carburetor
demand.
0
0
0.2
0.4
0.6
0.8
1.0
Carburetor demand (Dc)
6.3 FUEL INJECTION--SPARK IGNITION
Fuel Injection Systems
Spark ignition engines use fuel injectors to spray fuel into the air stream at the intake
manifold (throttle body injection), at the inlet port (port fuel injection), or directly into
the cylinder (direct-injection). Figure 6.4 shows an example of a system using port fuel
injection and Figure 6.5 shows the fuel injector in a direct-injection engine. With port fuel
injection, the fuel is sprayed into the port and onto the inlet valve to cool the valve and
begin vaporization of the fuel. The amount of fuel required can be large enough so that the
fuel injector will continue spraying into the port even when the valve is closed, so that the
fuel can not enter the cylinder until the next intake stroke. For increased power, some spark
ignition engines are configured with dual fuel injection, in which both port and direct fuel
injectors are used to deliver fuel to the cylinder.
Digitally controlled fuel injectors were first patented in 1970 and were used on production vehicles in the United States beginning in 1982. Most automotive engines use
port fuel injection, with direct fuel injection increasing in use. The required fuel pressure
depends on the location of the fuel injector. A port fuel injector will have a fuel pres-
Fuel injector
Fuel
spray
Figure 6.4 Schematic of port
fuel injection.
Intake flow
Fuel Injection--Spark Ignition
167
Fuel injector
Fuel spray
Piston bowl
Figure 6.5 Schematic of gasoline direct fuel
injection. (Adapted from Takagi et al., 1998.)
sure of about 2--5 bar, and a direct fuel injector will have a fuel pressure on the order of
30--130 bar.
Spark ignition engines primarily use a pintle nozzle fuel injector, where the upward
motion of a pintle nozzle opens the valve. At an appropriate time in the engine cycle, the
engine control computer issues a square wave pulse to open and close the pintle nozzle. The
pintle is rapidly lifted off its seat by a solenoid and the quantity of fuel injected increases
more or less linearly with the duration of the open period, since the opening and closing
times are much less than the open duration time. Direct fuel injectors employ swirl ports
to induce a tangential motion to the spray, resulting in a hollow cone spray for enhanced
atomization and vaporization.
Bernoulli’s equation can be used to estimate the mass of fuel injected as a function
of the pressure and open period. Assuming quasi-steady flow through the fuel nozzle, the
mass of fuel injected in one period Δ𝑡 is the integral of the fuel flow rate:
𝑚f =
∫0
Δ𝑡
𝑚̇ f 𝑑𝑡 = (2𝜌f Δ𝑃 )1∕2
∫0
Δ𝑡
𝐴f 𝑑𝑡
(6.7)
where Δ𝑃 is the difference between the fuel delivery pressure and the air pressure upstream
of the throttle, and 𝐴f is the time-dependent nozzle effective area. The average effective
flow area 𝐴̄ f of the injector nozzle is given in Equation 6.8:
1
𝐴̄ f =
Δ𝑡 ∫0
Δ𝑡
𝐴f 𝑑𝑡
(6.8)
Defining the average injector discharge coefficient 𝐶̄d as
𝐶̄d =
𝐴̄ f
𝐴o
(6.9)
where 𝐴o is the total geometric orifice area, the mass of fuel injected during the open
duration period Δ𝑡 can be expressed as
𝑚f = (2𝜌f Δ𝑃 )1∕2 𝐶̄d 𝐴o Δ𝑡
(6.10)
168
Fuel and Airflow in the Cylinder
Fuel per pulse (mg)
40
Injection pressure
(MPa)
30
200
80
20
20
10
Figure 6.6 Mass of fuel injected as
a function of injector pulse width
and pressure.
0
0
0.2
0.4
0.6
0.8
1.0
1.2
Pulse width (ms)
The open duration period Δ𝑡 of the fuel injection is typically 1--10 ms. The fuel per pulse
is plotted in Figure 6.6 as a function of the pulse width and pressure for a representative
automotive fuel injector. The prediction that the mass of fuel injected is linearly proportional
to the open duration is borne out by experiment as evidenced by the results shown in
Figure 6.6.
It is important that the fuel mostly evaporate before delivery to the cylinder. If the fuel
in the air were to exist as large droplets, then these would collide with the intake manifold
walls, and form a liquid fuel film on the walls. Accumulation of liquid fuel on the walls
alters in an uncontrolled manner the fuel--air ratio of the fuel delivered to the cylinders.
It causes lags and overshoots in the fuel flow with respect to the airflow delivered, and it
causes variations in the fuel--air ratio from cylinder to cylinder.
Fuel injection in natural gas engines is discussed in Kim et al. (2004). Due to the
relatively high-pressures involved, consideration needs to be given to the compressible
flow features such as Mach disks that occur with natural gas fuel injection. For further
information, the reader is referred to Li et al. (2004).
6.4 FUEL INJECTION--COMPRESSION IGNITION
Diesel Injection Systems
With compression ignition engines, fuel injection is classified as either direct-injection in
which fuel is sprayed directly into the cylinders or indirect-injection (IDI), in which fuel
is sprayed into a prechamber connected to the main chamber. One of the limiting features
of compression ignition engines is the finite mixing rate of the fuel and air, especially for
higher speed diesel engines, so a variety of DI and IDI systems have been developed to
rapidly form a combustible fuel--air mixture. Fuel injection systems in the early parts of the
twentieth century used air injection, a technique using the Venturi effect where compressed
air entrained diesel fuel in the injector and carried it into the cylinder.
In direct-injection, the fuel is injected into the cylinder after the intake air has been
compressed to about 50 bar and a temperature high enough (>850 K) for autoignition
of the fuel--air mixture. The fuel flows through very small orifices in the injector tip,
forming a liquid jet that subsequently breaks up into droplets that evaporate and mix with
the surrounding air. Direct-injection systems operate at high-pressures, on the order of
Fuel Injection--Compression Ignition
169
1000 bar, so the fuel velocity will be high enough to penetrate deeply into the cylinder, and
the atomized droplets will be small enough for rapid evaporation and subsequent ignition
during the time available before the piston reaches top dead center. Distribution of the fuel
in the cylinder is accomplished by both penetration along the streamwise flow direction, and
dispersion perpendicular to the flow. The relative amounts of penetration and dispersion
needed depend on the cylinder and piston geometry and the location of fuel injection.
We can again use Bernoulli’s equation to examine an engine speed--injection pressure
issue that arises with fuel injection systems. Assuming quasi-steady flow of an incompressible fluid, the total mass of fuel injected into a cylinder is
𝑚f = (2𝜌f Δ𝑃 )1∕2 𝐴̄ f
Δ𝜃 1
2𝜋 𝑁
(6.11)
Note that this expression is identical to Equation 6.10 except that Δ𝑡 is expressed in terms
of the crank angle change during the injection duration.
It is clear, in this form, that in order to hold Δ𝜃 constant as engine speed varies, one
must accordingly increase or decrease the fuel pressure to hold 𝑚f constant. In fact, since
typically the fuel injection pressure is large compared with the cylinder pressure, one must
vary the fuel injection pressure 𝑃f with the square of engine speed:
𝑃f ≈ Δ𝑃 ∝ 𝑁 2
if 𝑚f , Δ𝜃 are constant
(6.12)
Herein lies one basic problem with trying to build a diesel engine that will operate
over a large speed range: if 𝑁max ∕𝑁min = 5, then 𝑃f, max ∕𝑃f, min = 25; furthermore, if at
low speeds 𝑃f, min = 50 bar is needed to ensure good atomization and penetration into
the combustion chamber, then 𝑃f, max = 1250 bar. This issue can be addressed with highpressure fuel pumps.
Fuel injection systems are generally classified into two categories: systems that separate the fuel pressurization and injection, and those that combine fuel pressurization and
injection.
The common rail injection system is an example of a system that separates fuel pressurization and injection. The high-pressure required for injection is generated mechanically
using one common high-pressure pump. The high-pressure fuel is contained inside a thickwalled tube called a common rail. A control valve allows the fuel pressure to be maintained
at a level set by the engine control unit. The fuel rail is large enough so that the internal
pressure is not affected by operation of the fuel injectors. Since the fuel pressure is maintained at a constant value, this injection system is capable of multiple (pre-, main, and
post-) injections for reduction of emissions and noise.
A schematic of a mechanically controlled common rail fuel injector is shown in
Figure 6.7, and an electrically activated common rail fuel injector is shown schematically
in Figure 6.8. In Figure 6.8, the electrically activated solenoid controls the motion of a
control ball valve that regulates the flow of fuel from a valve control chamber. A needle
valve is lifted, opening a flow path through the nozzle, and the fuel, which is already at a
high-pressure, is injected into the engine cylinder. Excess fuel flows by the ball valve and
back to the lower pressure fuel tank.
Examples of systems that combine the fuel pressurization and injection process are
unit pumps and unit injectors. In these systems, there is simultaneous pressure generation
and injection. The pressure generation is initiated by a camshaft or electric solenoid. These
systems can have greater peak injection pressures than common rail systems, as the shape
of the cam controls the pressure profile. Since the camshaft is coupled to the engine, the
maximum injection pressure increases with engine speed.
170
Fuel and Airflow in the Cylinder
Lifter
Leak return
Needle valve
High-pressure supply
Figure 6.7 Common rail fuel
injector--mechanical control.
Solenoid
Fuel
return
Ball
valve
High
pressure
supply
Valve
control
chamber
Control
plunger
Nozzle
chamber
Figure 6.8 Common rail fuel
injector--electrical control.
Spray
171
Fuel Injection--Compression Ignition
High-pressure line
Check
valve
Inlet port
Lowpressure
supply
Pumping
plunger
Nozzle
Cam
Figure 6.9
system.
Unit pump fuel injection
The unit pump systems utilize the principle depicted in Figure 6.9. A low-pressure
transfer pump fills the cavity ahead of a pumping plunger. A cam is configured to displace
the plunger at the time when injection is to occur. The plunger moves up, shuts off the
inlet port, and, because the fuel is nearly incompressible, it rapidly increases the fuel
pressure. The rise in fuel pressure creates a pressure imbalance on the needle in the injector
nozzle, causing it to open and allowing fuel to discharge into the engine cylinder through
the nozzle. Once the fuel pressure falls to some predetermined value, a spring forces the
needle down shutting off the injector. Typically, the injector pressure is about 500 bar
when the needle opens, and it increases to a maximum of about 850 bar just before the
needle closes.
With a unit pump injection system, the mass of fuel injected is controlled by varying
the displacement of the pumping plunger. One way in which this is done is shown in
Figure 6.10. A helix is cut into the pumping plunger that reopens the inlet port at some
intermediate position in the plungers stroke. A rack and pinion arrangement varies the
effective stroke by rotating the plunger and therefore the position at which the port will
reopen and thus dropping the fuel pressure. Some older diesel injection systems use a
positive displacement pump so that the mass injected is the independent variable and the
fuel pressure adjusts itself accordingly.
A unit injector is a combined unit consisting of both the pump and injector. Unit
injectors were invented in 1934, and for many years were mechanically controlled. In
Pump barrel
Pump plunger
Figure 6.10 Unit pump
operation.
Zero delivery
Inlet port
Effective stroke
Partial delivery
Helix
Rack rod
Maximum delivery
172
Fuel and Airflow in the Cylinder
1995, electronic unit injectors were developed. Metering is accomplished by actuation of
a solenoid-operated valve, and closure of the solenoid valve initiates pressurization and
subsequent fuel injection. The duration of valve actuation determines the amount of fuel
injected. Unit injectors are commonly used on locomotive diesel engines.
Various nozzle configurations are used for diesel spray injection, including pintle,
single, and multiorifice. Needle valves are used in diesel injection systems to control the
amount of fuel injected. The sac volume, the volume of fuel in the space between the
needle and the orifice is designed to be as small as possible to reduce unwanted fuel
injection. Typical nozzle diameters 𝑑n are on the order of 200 μm, and length 𝐿n about 1 mm.
EXAMPLE 6.1
Diesel Fuel Injection
The specifications for a 12-cylinder four-stroke diesel engine being designed are that it
operate at a speed of 𝑁 = 1200 rpm and produce 𝑊̇ b = 500 kW of power, with a brake
specific fuel consumption (bsfc) of 0.25 kg/kWh. The cylinder pressure at the start of
injection is 30 bar, and the maximum cylinder pressure during combustion is 60 bar. The
injection duration is nominally 10◦ of the crank angle. The unit pump injector nozzle is
set to open at 200 bar, with a maximum injector pressure of 600 bar. The injector has a
coefficient of discharge 𝐶̄d = 0.60.
(a) What is the mass of fuel injected per cylinder per cycle?
(b) What total orifice area 𝐴o for each injector should be selected?
SOLUTION (a) The cycle average fuel consumption rate per cylinder is
𝑚̄̇f = bsf c ⋅ 𝑊̇ b ∕𝑛c = 0.25 ⋅ 500∕(12 ⋅ 60) = 0.174 kg∕min
so the mass of fuel injected per cylinder per cycle is
𝑚f = 𝑚̄̇f ∕(𝑁∕2) = 0.174∕(1200∕2) = 2.89 × 10−4 kg
(b) The pressure difference at the beginning of injection is 200 − 30 = 170 bar, and at
the end of injection is 600 − 60 = 540 bar. For this preliminary design, let us estimate an
average pressure difference of (540 + 170) / 2 = 355 bar between the fuel injector and
the cylinder. Using Equation 6.11, and assuming the diesel fuel is incompressible with a
density 𝜌f = 840 kg/m3 ,
𝐴o =
=
𝑚f (2𝜌f Δ𝑃 )−1∕2
(
)
Δ𝜃
⋅ 𝑁1
𝐶̄d 360
2.89 × 10−4 (2 ⋅ 840 ⋅ 355 × 105 )−1∕2
)
(
10
60
0.60 360
⋅ 1200
= 1.42 × 10−6 m2 = 1.42 mm2
Diesel Sprays
The formation of a combustible mixture of diesel fuel and air in an engine cylinder is a very
complex two-phase fluid mechanics process. Diesel fuel injection has a number of major
Fuel Injection--Compression Ignition
173
features, including an initial spreading angle, entrainment of surrounding gas, a liquid core
surrounded by a vapor sheath, jet atomization into droplets, droplet evaporation, and gas
jet wall impingement.
The spray is initially conical in shape, with the virtual origin of the jet inside the nozzle.
The spreading angle of the spray, 𝜃, depends on the ratio of the cylinder gas and diesel fuel
densities. Experimental data in the atomization regime has been correlated by Reitz and
Bracco (1979) with the following expression:
( )1∕2 √
3
4𝜋 𝜌g
𝜃
(6.13)
tan =
2
𝐴 𝜌l
6
where 𝜌g is the gas density and the parameter 𝐴 is a function of the nozzle diameter 𝑑n and
length 𝐿n :
𝐴 = 3.0 + 0.28(𝐿n ∕𝑑n )
(6.14)
For example, for values of 𝜌g ∕𝜌l = 25 × 10−3 , and 𝐴 = 5, the spread angle 𝜃 = 13◦ . As
the gas density increases, the spreading angle of the spray increases.
There are different jet breakup mechanisms, depending on the outlet jet velocity and
physical properties. The major governing parameters are the Weber number, the ratio of
the shear and the surface tension forces; and the Reynolds number, the ratio of the inertial
and the viscous forces:
𝑊 𝑒 = 𝜌l 𝑢2 𝑑n ∕𝜎
(6.15)
𝑅𝑒 = 𝜌l 𝑢𝑑n ∕𝜇l
(6.16)
where 𝜌l is the liquid jet density, 𝑢 is the relative velocity of the jet, 𝑑n is a nozzle or droplet
diameter, 𝜎 is the surface tension , and 𝜇l is the dynamic viscosity of the jet.
Due to the high injection pressures, the outlet jet relative velocity is on the order of
100--200 m/s, which is in an atomization breakup regime. The shear forces on the liquid
jet from the surrounding gas result in unstable surface waves that pinch off the liquid jet,
producing droplets. In the atomization regime, the liquid fuel jet breaks up into liquid drops
and ligaments, with dimensions much less than the injection nozzle diameter. With high
injection pressures or short nozzles, the breakup length decreases, and breakup can occur
at the nozzle exit. With the decrease in static pressure in the nozzle accompanying the fluid
acceleration, it is also possible for cavitation to occur inside the nozzle, producing a bubbly
two-phase injection flow from the nozzle. Collisions between droplets can also result in
droplet coalescence.
The spray penetration distance has been the subject of extensive experimental and
computational research due to its importance in the behavior of subsequent combustion
processes. If the penetration is too short, air at the edges of the cylinder will not be involved
in the combustion process, and if the penetration is too long, the jet spray will impinge on
the cylinder walls, which will reduce the jet velocity and entrainment. Depending on the
drop Weber number, the impact of a drop on the cylinder wall can result in the formation
of a liquid film on the surface, breakup and vaporization, or rebound back into the cylinder.
A widely used correlation by Dent (1971) for the spray penetration distance is
)1∕4
)1∕4
(
(
294
Δ𝑃
(𝑡𝑑n )1∕2
(6.17)
𝑆 = 3.07
𝜌g
𝑇g
where Δ𝑃 is the injector nozzle pressure drop, 𝑡 is the time after start of injection. All
the parameters in the above equation are in SI units. As the injection pressure or the nozzle
diameter is increased, the penetration distance is increased. Once the startup phase of the
174
Fuel and Airflow in the Cylinder
injection is complete, the length of the liquid core remains fairly constant until the end of
injection.
The droplet sizes are characterized by a Sauter mean diameter, 𝐷SM , named after a
German scientist, J. Sauter, who in 1928 developed a measure of average particle size
for a distribution of particles. The Sauter mean diameter, 𝐷SM , is defined as the diameter
of a model drop whose volume to surface ratio is equal to that of the total spray. A
representative value of 𝐷SM for diesel spray is 50 microns. As the gas density and nozzle
diameter decrease, 𝐷SM decreases, increasing the surface--volume ratio. Collisions between
droplets also change the droplet size distribution.
∑𝑛
3
i=1 𝑑
(6.18)
𝐷SM = ∑𝑛 𝑖
2
i=1 𝑑𝑖
The next step in the diesel fuel injection process is the evaporation of the droplets. The
evaporation time, on the order of 1 ms, depends on the droplet size, velocity, temperature,
and droplet interaction. The energy for evaporation is heat transfer from the surrounding
hot gases, an adiabatic saturation process that raises the droplet surface temperature to the
saturation temperature, and lowers the surrounding gas temperature.
As mentioned earlier, the diesel combustion process is mixing limited. At the start
of combustion, about 80% of the injected fuel has evaporated and is in the vapor phase.
However, only about 20% of the fuel vapor has mixed with the surrounding air sufficiently to
be flammable, so additional mixing is required for complete burning during the combustion
process. For further information about the modeling and computation of fuel--air mixing
processes in engines, the reader is referred to the book by Baumgarten (2006).
6.5 LARGE-SCALE IN-CYLINDER FLOW
Introduction
There are large-scale flow structures that are present in the cylinder during the intake and
compression stroke, and are accompanied by the generation of small-scale turbulence. In
this section, we discuss the physical and computational methods for quantification and
characterization of these large-scale flow structures. Three parameters that are used to
characterize the large-scale fluid motion and mixing in the cylinder are swirl, squish, and
tumble. With large-scale mixing, the characteristic length of the fluid motion is on the order
of the combustion chamber diameter, whereas with small-scale mixing, the turbulent fluid
vortex size is many orders of magnitude smaller.
Cylinder Flow Measurement Techniques
Using lasers, it has become possible to measure local instantaneous velocities, temperatures,
and some species concentrations within the cylinder without insertion of intrusive probes.
Research engines and test rigs are built with optical access for the laser as one of the
primary design features.
Figure 6.11 shows an arrangement for measuring velocity using a laser Doppler velocimetry (LDV) measurement technique. The arrangement shown is a steady-flow test rig.
The beam from an argon-ion laser is split into two beams that are then focused to a small
volume within the flow. Small particles, about 0.5 μm in diameter, are deliberately added
to the flow to track the gas speed. As these particles pass through the probe volume made
by the intersecting laser beams, they scatter radiation in all directions. The Doppler effect
Large-Scale in-Cylinder Flow
175
Window
Shifter
Photomultiplier
Ar laser
Mixer
Beam splitter
Lens
Figure 6.11 Laser
Doppler velocimetry
(LDV) steady-flow
test rig.
Lens
Tracker
Mean Velocity
velocity fluctuation
Cylinder
shifts the frequency of the scattered light. The frequency shift is proportional to the particle
velocity. The electronics of the LDV system filter and process the signal to detect the frequency shifts. Both the mean and turbulent velocity components are measured. By moving
the laser beams and thus the probe volume, the velocity can be measured at different points
within the cylinder.
Particle image velocimetry (PIV) systems measure velocity by determining particle
displacement over time using a double-pulsed laser technique. A pulsed laser light sheet
illuminates a plane in the flow, and the positions of particles in that plane are recorded by
a video camera. A fraction of a second later, another laser pulse illuminates the same plane
creating a second image. Images on the two planes are analyzed using cross-correlation
techniques to compute the turbulent velocity field. Additional information about PIV and
other laser-based measurement techniques is given in Adrian (1991).
Computational Simulation of In-Cylinder Flow Fields
The in-cylinder flow field can realistically be simulated using computational fluid dynamics
(CFD) analysis. There are a number of CFD programs such as VECTIS (VECTIS USERS
MANUAL, 2014), STAR-CD (STAR-CD USERS MANUAL, 2014), FLUENT (FLUENT
USERS MANUAL, 2014), and CONVERGE (CONVERGE 2.1 MANUAL, 2013) that are
available for computation of in-cylinder flow fields. These programs solve the discretized
Navier--Stokes equations, with user chosen turbulence models, on a three-dimensional mesh
or grid. The features and models included in current CFD codes include moving meshes,
injection, spray and droplet evaporation, and turbulent combustion. Postprocessing is used
for the analysis and visualization of the resulting solution.
A representative CFD grid for a four valve SI engine is given in Figure 6.12, and
a close up cutaway of the port and valve region is shown in Figure 6.13. The computed
flow field at 120◦ after tdc during the intake stroke is shown in Figure 6.14. Note that the
computed flow field downstream of the intake valve is characterized by large-scale vortex
motion.
With continued advances in high-speed digital computation, CFD calculations are now
systematically included in internal combustion engine engineering design and optimization
processes. For further information about the use of CFD analysis in the engine design
process, the reader is referred to the book by Shi et al. (2011).
Swirl and Tumble
Swirl refers to a large-scale vortex motion within the cylinder about its long axis, and
tumble is a large-scale vortex motion perpendicular to the cylinder axis. Swirl is generated
during the intake stroke either by tangentially directing the flow into the cylinder using
176
Fuel and Airflow in the Cylinder
Figure 6.12 CFD grid for in-cylinder
flow of a four-valve cylinder.
(Courtesy Adapco.)
Figure 6.13 Close-up of CFD grid.
(Courtesy Adapco.)
Figure 6.14 CFD flow field.
(Courtesy Adapco.)
Large-Scale in-Cylinder Flow
177
O
r
Figure 6.15 Schematic of intake port showing swirl
parameters 𝑅v and 𝛼. Adapted from Uzkan et al. (1983).
directed ports or by preswirling the incoming flow by use of a helical port. Helical ports
are generally more compact than directed ports. They are capable of producing more swirl
than directed ports at low lifts, but are inferior at higher lifts. Either design creates swirl at
the expense of volumetric efficiency. Tumble is induced by the inlet poppet valve.
Swirl and tumble are one of the principal means to ensure rapid mixing between fuel
and air in direct-injected engines. In diesel engines, as fuel is injected, the swirl bends
the fuel jet and convects it away from the fuel injector, making fresh air available for the
following fuel upstream. Swirl and tumble are also used in gasoline engines to promote
rapid combustion, as they will result in higher turbulence levels at the start of ignition.
The swirl and tumble generated during the intake stroke will decay due to wall friction
and turbulent dissipation. The swirl level at the end of the compression process is dependent
upon the initial swirl generated during the intake process and how much it is amplified
during the compression process.
Some parameters to consider in the design of a port for swirl are shown in Figure 6.15.
These are the radius of the valve offset 𝑅v and the orientation angle 𝛼. Research and
development work, like that for maximizing the discharge coefficient, is typically done on
a steady-flow bench. One way in which the swirl produced can be measured is shown in
Figure 6.16. A honeycomb structure of low mass, supported by a low-friction air bearing
straightens the flow. The change in angular momentum of the flow applies a torque to the
honeycomb, which is measured by recording the force required to restrain it. The swirl is
proportional to that torque.
The efficiency of the port as a swirl producer is characterized by a swirl coefficient 𝐶s
defined in Equation 6.19 as
̇ 𝑏∕2)
𝐶s = 𝜏∕(𝑚𝑈
(6.19)
where
𝜏
𝑚̇
𝑈
𝑏
= torque applied to honeycomb
= mass flow rate
= discharge velocity of gas
= cylinder bore
The swirl coefficient 𝐶s is equal to 1 for the limiting case where the inlet flow enters
tangentially at the cylinder wall. 𝐶s increases with the valve lift or offset, and the port
178
Fuel and Airflow in the Cylinder
Air flow
Swirl
Honeycomb
Torque
Figure 6.16 Steady-state flow and swirl system.
Adapted from Uzkan et al.(1983).
Air bearing
orientation is important only at the larger lifts. At zero offset, the port is producing swirl
because of the helical path upstream of the valve. Since the swirl coefficient 𝐶s characterizes
the overall angular momentum of the flow, it does not capture all the complexity of the
inlet flow, as it is possible for many different velocity distributions within the cylinder to
yield the same angular momentum.
A bowl within the piston crown or cylinder head can be used to amplify swirl during
the compression stroke, as shown in Figure 6.17. The swirl ratio in Figure 6.18 varies from
zero to six times the engine speed. The swirl is proportional to the angular momentum,
but it is also inversely proportional to the moment of inertia. At top center, the moment
of inertia goes through a minimum in a manner dependent upon the design of the piston
bowl. As seen in Figure 6.18, near top dead center of compression the swirl increases and
decreases in a rather short period. The deeper the bowl, at constant compression ratio, the
greater is the change in the moment of inertia and the greater is the swirl amplification.
b
1
2
Vcup
3
d
Figure 6.17 Schematic of bowl in piston crown
for production of swirl and squish.
1
h
Large-Scale in-Cylinder Flow
179
8
Squish bip
Swirl ratio
6
3
2
Figure 6.18 Example plot of swirl ratio versus
crank angle. Adapted from Belaire et al. (1983).
0
–400
–200
0
200
Crank angle
400
In operating engines, a swirl ratio 𝑅s is used to characterize the swirl:
𝑅s = 𝜔s ∕2𝜋𝑁
(6.20)
and similarly, a tumble ratio 𝑅t is used to characterize the tumble:
𝑅t = 𝜔t ∕2𝜋𝑁
(6.21)
The swirl and tumble ratios are defined as the ratio of the solid body parallel and
perpendicular rotational speeds of the intake flow 𝜔s and 𝜔t to the engine speed 2𝜋𝑁. The
solid body rotational speed is defined to have the same angular momentum as the actual
flow. Additional discussion about the role of swirl and tumble in engines is contained in
Kajiyama et al. (1984), Kawashima et al. (1998), and Lumley (1999).
There are limits to the amount of swirl that can be used effectively to minimize demands
on the fuel injection system. Herein lies one of the primary reasons for building divided
chamber, or, as they are often called, indirect-injection engines; less reliance on air motion
induced by the fuel injection is required to effect large-scale mixing. A prechamber (Olsen
and Kirkpatrick, 2008) used in large-bore natural gas fueled engines to increase in-cylinder
fluid motion is shown in Figure 6.19.
With a prechamber or swirl chamber, air is forced to flow into the chamber during the
compression stroke establishing three-dimensional air motion and generating turbulence.
Figure 6.19 Prechamber for use in
large-bore natural gas engine.
180
Fuel and Airflow in the Cylinder
The pressure rise during combustion in the prechamber creates a flow out of the prechamber
and back into the cylinder. The velocities of that backflow can be rather high, creating
turbulence at the expense of an additional pressure loss. Flow passages are often contained
within the piston top to organize the back flow into the cylinder to create large-scale mixing
of the combustion products and the cylinder air.
Squish
Squish is a radial flow occurring at the end of the compression stroke in which the compressed gases flow into a cup located within the piston or a wedge in the cylinder head. The
squish flow results from the cup-shaped geometry. The amount of squish is defined by the
relative squish velocity. Incorporation of a bowl into the piston not only amplifies swirl,
but also induces squish.
This can be appreciated in terms of a rather simple argument based on the continuity
equation, and shown in Figure 6.17. The density within the cylinder at any time is more or
less uniform (though time-dependent) during the compression stroke. Thus, at any instant,
the mass within any of the zones labeled (1), (2), and (3) is proportional to the volume in
these zones at any time. During compression, zones (1) and (2) get smaller, whereas zone
(3) remains fixed. Thus, during compression, mass must flow out of zones (1) and (2), into
zone (3). The velocity of the gas crossing the control surface between zones (1) and (2) is
called the squish velocity and zone (1) is called the squish zone.
Use of squish was pioneered by H. Ricardo, in order to increase the turbulence level in
side-valve engines, which were prevalent in the first half of the twentieth century (Lumley,
2001). In modern open chamber four-valve pent roof engines, the squish area is relatively
low.
6.6 IN-CYLINDER TURBULENCE
Turbulence Parameters
The Reynolds numbers of flows in engine cylinders are on the order of 10,000--50,000,
well into the turbulent flow regime. The turbulence results from the high-velocity inlet
flow from the intake valve or port into the cylinder during the intake stroke. The inlet fluid
jet flows across the cylinder, impinges on the piston top and cylinder walls, creating both
large- and small-scale fluid flow features. The impinging flow is composed of turbulent
eddies that have lifetimes comparable to the intake stroke timescale.
Turbulent flow in an engine can be envisioned as a mean fluid flow upon which
are superimposed vortices of different sizes randomly dispersed in the flow. Turbulence
is inherently three dimensional and time-dependent. A turbulent flow is composed of
numerous vortices or eddies that have finite lifetimes and appear to be born at random times.
The axes of the vortices also assume random orientations. There are even vortices within
vortices. The turbulence in an engine is of importance as it controls the rate of combustion,
since the combustion flame front is convected across the cylinder by turbulent vortices.
The turbulence in the flow field begins to appear above critical values, about 2300, of
the mean flow Reynolds number, a ratio of the inertial to the viscous stresses. The Reynolds
number is named after Osborne Reynolds (1842--1912), an English engineering professor
who proposed it in 1883. The mean flow Reynolds number is defined in Equation 6.22,
with the cylinder bore 𝑏 as a length scale, and the mean piston speed 𝑈̄ p as a velocity scale:
𝑅𝑒 = 𝑈̄ p 𝑏∕𝜈
(6.22)
In-Cylinder Turbulence
181
It is not until the flow is analyzed statistically that any regularity in the flow field begins
to appear. Flows that are statistically periodic, as in the case with reciprocating internal
combustion engines, are treated using a statistical procedure called ensemble averaging.
The ensemble average velocity 𝑈̄ (𝑥, 𝜃) is defined as
𝑛
1∑
𝑈 (𝑥, 𝜃, 𝑗)
𝑈̄ (𝑥, 𝜃) =
𝑛 j=1
(6.23)
where 𝑛 is the number of cycles averaged and 𝜃 varies from 0 to 4𝜋 for a four-stroke engine
and from 0 to 2𝜋 for a two-stroke engine. The left-hand side of Equation 6.23 is read as the
ensemble average of the velocity at position 𝑥 within the flow and at a time corresponding
to the crank angle 𝜃 . The velocity summed on the right-hand side is the velocity at position
𝑥 and angle 𝜃 for the jth cycle. As a consequence of the cycle-by-cycle variation in average
velocity, there is a difference 𝑈̂ (𝑥, 𝜃, 𝑗) between the average velocity at a given location
for a given cycle 𝑗 and the ensemble average velocity:
𝑈̂ (𝑥, 𝜃, 𝑗) = 𝑈̄ (𝑥, 𝜃, 𝑗) − 𝑈̄ (𝑥, 𝜃)
(6.24)
To define the instantaneous turbulence within a given cycle 𝑗, one writes 6.25
𝑈 (𝑥, 𝜃, 𝑗) = 𝑈̄ (𝑥, 𝜃) + 𝑢′ (𝑥, 𝜃, 𝑗)
(6.25)
𝑢′
where
is the turbulent fluctuation relative to the ensemble average, that is, the difference between the ensemble average and the instantaneous velocity, not the difference
between a given cycle average and the instantaneous velocity. To quantify the magnitude
of the turbulent fluctuations, a root-mean-square turbulence intensity 𝑢t (𝑥, 𝜃) is defined in
Equation 6.26, using ensemble averaging:
[ 𝑛
]1∕2
1 ∑ ′2
(6.26)
𝑢 (𝑥, 𝜃, 𝑗)
𝑢t (𝑥, 𝜃) =
𝑛 j=1
The kinetic energy per unit mass of the turbulent fluctuations is
1 ′̄ ′ 3 2
𝑢𝑢 ≈ 𝑢
(6.27)
2 𝑖 𝑖 2 t
Determination of the turbulence intensity 𝑢t (𝑥, 𝜃) requires measurements of 𝑈 (𝑥, 𝜃, 𝑗)
and 𝑢′ as a function of position and crank angle. Turbulence characteristics of flows in
engine cylinders have been measured using both hot-wire anemometry and laser Doppler
velocimetry. Since the inlet jet velocity is proportional to piston speed, one would expect
the in-cylinder mean and fluctuating velocities to also be proportional to piston speed,
which then can be used as a normalizing parameter.
The results from experiments indicate that the turbulence intensity varies a great deal
over a cycle. The maximum value of the turbulence intensity normalized by the piston
speed, 𝑢t ∕𝑈̄ p , is about ten, and occurs at 90◦ after top dead center, that is, halfway down
the intake stroke, which is near the location of the maximum piston speed. The normalized
turbulence intensity decreases to about one at bottom dead center, and remains at a value
of order one during the compression stroke, and is almost homogeneous.
One of the most important conclusions reached to date is that the turbulence intensity
increases linearly with piston speed. Liou, Hall, Santavicca, and Bracco (1984) conclude
from a review of experimental results that the normalized top dead center turbulent intensity
is
𝑘=
𝑢t ∕𝑈̄ p ≈ 0.5
(6.28)
182
Fuel and Airflow in the Cylinder
Of course, there are differences from engine to engine at the same piston speed. The
differences are caused partly by differences in the engine design and partly because flow
cannot be quantitatively characterized by a measurement of only one velocity component
at just one point. The turbulence measurements cover a range of engine configurations
including engines with and without swirl. In the same engine with and without swirl, it has
been found that the turbulent intensity is increased by swirling the flow.
The turbulence in an engine cylinder is characterized by four length scales: the characteristic length 𝐿, the integral length scale 𝑙, the Taylor microscale 𝜆, and the Kolmogorov
microscale 𝜂.
The characteristic length 𝐿 of the enclosure represents the largest possible eddy size
that the confining geometry of the walls will allow, such as the cylinder bore or clearance
height. For a cylindrical combustion chamber, one should expect near top center, that the
characteristic length should be roughly equal to the clearance height ℎ; whereas near bottom
center, the characteristic length should be roughly equal to the bore 𝑏. With a cylindrical cup
in the piston, near top center the characteristic length would be roughly the cup diameter.
The integral length scale 𝑙, represents the size of the largest and thus most energetic
eddies in the turbulent flow field. For an inlet flow past a poppet valve, the integral scale
is roughly equal to the inlet jet thickness. The integral scale is defined as the distance
between two points where the autocorrelation coefficient of the fluctuating velocity at the
points goes to zero. A number of significant turbulence parameters are based on the integral
scale: The turbulence Reynolds number 𝑅𝑒t is based on the integral scale and the turbulent
velocity, as given by Equation 6.29:
𝑢t 𝑙
𝜈
The integral scale is related to the rate of dissipation 𝜖:
𝑅𝑒t =
𝜖=
𝑑𝑢2t
𝑑𝑡
∼
𝑢2t
𝜏l
=
𝑢3t
𝑙
(6.29)
(6.30)
and to the energy spectra 𝐸(𝜅)
𝑙=
𝜋
2𝑢2 ∫0
∞
𝐸(𝜅)
𝑑𝜅
𝜅
(6.31)
The turbulent eddy viscosity 𝜈t is the product of the integral scale and turbulent
velocity:
𝜈 t = 𝑢t 𝑙
(6.32)
The integral timescale, 𝜏l = 𝑙∕𝑢t , represents the lifetime of a turbulent eddy. The Taylor
microscale 𝜆 is useful in estimating the mean strain rate of the turbulence. It is defined as
( )2 𝑢2
𝜕𝑢
= t
(6.33)
𝜕𝑥
𝜆2
The Kolmogorov microscale 𝜂 is the smallest size viscous damping will allow. The
Kolmogorov microscale is named after A. Kolmogorov (1903--1987), a Russian mathematician, who proposed that the smallest scales of turbulence are universal and depend only
on the dissipation rate 𝜖 and viscosity 𝜈. For turbulent flow in internal combustion engines,
this scale is on the order of a few microns. From dimensional analysis, the Kolmogorov
microscale is
( 3 )1∕4
𝜈
(6.34)
𝜂=
𝜖
In-Cylinder Turbulence
183
Dimensional analysis of simple turbulent flows leads to the following relationships
between the four length scales:
𝑙 = 𝐶l 𝐿
( )1∕2
𝜆
15
−1∕2
𝑅𝑒t
=
𝑙
𝐶𝜆
( )−1∕4
𝜂
15
−3∕4
𝑅𝑒t
=
𝑙
𝐶𝜂
(6.35)
(6.36)
(6.37)
The constants 𝐶l , 𝐶𝜆 , and 𝐶𝜂 are numbers unique to the flow of interest and whose order
of magnitude is unity.
Thus, we see that if the integral scale can be determined, so can the other scales. Note
that the ratio of the largest to the smallest length scale is proportional to the Reynolds
number raised to the 3/4 power. For example, if the Reynolds number of a flow is 104 , then
𝐿∕𝑙 scales as 103 , and since turbulence is three dimensional, one would need about 109
grid points to resolve the entire range of length scales for each dependent variable. This has
implications for the numerical solution of turbulent flows in engines and will be discussed
in more detail in the next section.
As the turbulent Reynolds number increases, the smaller microscales decrease in size
according to Equations 6.36 and 6.37. Since the turbulence in an engine increases with
piston speed and the integral scale is independent of engine speed, we should expect that as
engine speed goes up, the microscales of the turbulence will go down. Experiments clearly
show that the flame wrinkling due to turbulence increases as the engine speed increases.
To fully characterize a turbulent flow, one needs to also specify the size distribution
of the vortices and eddies that make up the turbulence. The largest eddies are generated by
shear in the mean flow, and account for most of the transport of momentum and energy.
Inertial effects spread the turbulent energy from the large eddies to smaller and smaller
eddies until the viscous stresses are comparable to the inertial forces, a process called an
energy cascade. The timescales of the eddies scale with the size of the eddies, so the rate
at which energy is dissipated in the small eddies is controlled by the rate at which energy
is transferred from the large eddies to the small eddies. Measurements of the eddy energy
distribution indicate that the most energetic eddies initially have a size of about 1/6 of the
largest eddy (Townsend, 1976).
EXAMPLE 6.2
Turbulence Length Scales
An engine has a mean piston speed 𝑈̄ p of 5.0 m/s and a clearance volume height ℎ of 10 mm.
What is the characteristic length 𝐿, integral scale 𝑙, Taylor microscale 𝜆, and Kolmogorov
microscale 𝜂 at the end of compression? Assume the fluid kinematic viscosity at the end of
compression is 100 × 10−7 m2 /s and 𝐶𝜂 = 𝐶𝜆 = 1, 𝐶l = 0.2
SOLUTION 𝐿 = ℎ since the flow is constrained by the clearance volume geometry
𝐿 = 10 mm
𝑙 = 𝐶l 𝐿 = (0.2)(10) = 2 mm
( )1∕2
15
𝜆
−1∕2
𝑅𝑒t
=
𝑙
𝐶𝜆
𝑢t = 𝑈̄p ∕2 = 2.5 m∕s
184
Fuel and Airflow in the Cylinder
𝑢t 𝑙
(2.5)(2 × 10−3 )
= 500
=
𝜈
(100 × 10−7 )
( )1∕2
15
𝜆=
(500)−1∕2 (2) = 0.30 mm
1
𝑅𝑒t =
−3∕4
𝜂 = (𝐶𝜂 )−1∕4 𝑅𝑒t
𝑙
= (1)−1∕4 (500)−3∕4 (2) = 0.018 mm = 18 microns
Turbulence Models
Turbulence models have been developed so that statistical approximations of the exact
governing equations can be used. Due to the complexity of turbulent flow, turbulence
modeling has been and will remain an active research area. The validation process of
turbulence models is ongoing, and there are significant issues that need to be dealt with,
for example, specifying the initial conditions throughout the flow and boundary conditions
at the valves. In addition, the turbulence defined by Equation 6.26 does not recognize that
a part of the fluctuation may be due to cycle-to-cycle variation in an organized flow that in
any one cycle is different from the ensemble mean flow.
Turbulent flow fields in engines have been modeled for many years with a Reynoldsaveraged Navier--Stokes (RANS) turbulent models. There are a number of turbulent eddy
viscosity models currently being used by engine modelers. The most widely used is the
𝑘 − 𝜖 model. The 𝑘 − 𝜖 model is a two-equation model based on both a transport equation
for turbulent kinetic energy 𝑘, and a transport equation for the dissipation of turbulent
kinetic energy 𝜖 . The various forms of the 𝑘 − 𝜖 model assume that the turbulent eddy
viscosity depends on 𝑘 and 𝜖, as shown by the formulation given by Equations 6.38--6.40:
𝜈t = 𝐶𝜇 𝑘2 ∕𝜖
(
)
(
)
𝜈t 𝜕𝑘
𝜕 𝑈̄ i 𝜕 𝑈̄ i 𝜕 𝑈̄ j
Dk
𝜕
+
=
−𝜖
+ 𝜈t
Dt
𝜕𝑥j 𝜎k 𝜕𝑥j
𝜕𝑥j 𝜕𝑥j
𝜕𝑥i
(
)
(
)
̄
𝜈t 𝜕𝜖
D𝜖
𝜖2
𝜖 𝜕 𝑈̄ i 𝜕 𝑈̄ i 𝜕 𝑈j
𝜕
+
− 𝐶2
=
+ 𝐶1 𝜈t
Dt
𝜕𝑥j 𝜎𝜖 𝜕𝑥j
𝑘 𝜕𝑥j 𝜕𝑥j
𝜕𝑥i
𝑘
(6.38)
(6.39)
(6.40)
The constants 𝜎k , 𝜎𝑒 , 𝐶𝜇 , 𝐶1 , and 𝐶2 are empirical constants that are flow field dependent. The 𝑘 − 𝜖 model is based on a scalar eddy viscosity, so it does not take into account
nonisotropic effects on the turbulence field such as streamline curvature resulting from
cylinder swirl and tumble. It also assumes a fully developed turbulent flow field. Increased
accuracy of in-cylinder mixing computations, prediction of the turbulence level, and the
corresponding reaction rate can be accomplished by use of a compressible renormalized
group (RNG) 𝑘 − 𝜖 model, as discussed by Han and Reitz (1995).
Equations 6.38--6.40 are combined with the continuity, momentum, and energy equations to form a complete system for numerical analysis.
Large eddy simulation (LES) is a turbulence modeling procedure in which the large
eddies are computed, and the smallest eddies are modeled. LES turbulence models are more
computationally intensive than RANS models. The smallest eddies are more amenable to
modeling as they have greater isotropic turbulence characteristics than the larger eddies.
Eddies of size less than the grid size are removed from the dynamics. The grid cells can
Airflow in Two-Stroke Engines
185
be much larger than the Kolmogorov length scale. With LES modeling the time-dependent
Navier--Stokes equations are spatially filtered over the computational grid.
The LES decomposition is written as
𝑢(𝑥, 𝑡) = 𝑢(𝑥, 𝑡) + 𝑢′′ (𝑥, 𝑡)
(6.41)
In this decomposition 𝑢 is usually termed the large- or resolved-scale part of the solution,
and 𝑢′′ is called the small-scale or modeled part. It is important to note that both the large
and small-scales depend on both space and time, as opposed to Reynolds averaging where
only the fluctuating velocity component is time-dependent.
Both RANS and LES models are similar in that they both average over the smallscales, with the LES grid scales much smaller than the RANS grid scales. LES models are
available in the CFD computer programs mentioned previously.
Direct numerical simulation (DNS) resolves the entire range of turbulent length scales.
It is a complete time-dependent solution of the Navier--Stokes and continuity equations.
Since no turbulence model is used at any length scale, the grid must be small enough to
resolve the smallest turbulent eddy whose size is on the order of the Kolmogorov length
scale. The main advantage of LES over DNS is the much smaller computational expense,
as flow fields can be computed using LES at Reynolds numbers much higher than currently
practical with DNS.
For further information, the reader is referred to the turbulence modeling text by
Wilcox (2006). Garnier et al. (2009) also give further details about various aspects of LES
modeling, especially for compressible flows.
6.7 AIRFLOW IN TWO-STROKE ENGINES
Two-Stroke Scavenging Configurations
The two-stroke engine combines the intake and compression stroke and the expansion and
exhaust stroke in order to produce power every downward stroke. Two-stroke engines can
be either spark or compression ignition. A large number of different two-stroke engine
configurations have been designed, each with different scavenging or air-charging characteristics, airflow paths, and valve arrangements. In addition, there are many different valve
arrangements used to open and close ports, including piston control, poppet valves, rotary
valves, or sleeve valves. Considering the large number of possible permutations, based on
classification of the pumping method, the air path, and the valving arrangement, it is clear
why so many different types of two-stroke engines exist.
A crankcase-scavenged engine was discussed in Chapter 1. Air is inducted into the
crankcase, subsequently compressed, and pumped into the cylinder. Lubricating oil is added
to the intake airflow to lubricate the interior surfaces. Another class of two-stroke engines
is the separately scavenged engine in which a separate compressor, driven by the crank
or perhaps an exhaust turbine, delivers the air. Since there is no separate exhaust stroke,
the scavenging process relies on air being forced at elevated pressures into the cylinder to
expel the burnt exhaust gas from the previous cycle.
Two-stroke engines are classified on the basis of the air path during the course of
scavenging. The three scavenging configurations are cross-, loop-, and uniflow-scavenging.
Figure 6.20 illustrates various ways in which these three types of scavenging geometries
can be realized.
The first two-stroke engines designed by Dugald Clerk in the late 1800’s used cross
scavenging and a piston with a deflector top. With cross scavenging, the intake and exhaust
ports are located on opposite sides of the cylinder, 180◦ apart, and a deflector on the top of
186
Fuel and Airflow in the Cylinder
Cross scavenge
Figure 6.20 Two-stroke
scavenging configurations
(Taylor, 1985).
Through scavenge
with poppet
exhaust valves
Cross scavenge
with rotary
exhaust valve
Through scavenge
with sleeve
exhaust valves
Loop scavenge
Through scavenge
via opposed
pistons
the piston ideally directs the incoming scavenging flow upward toward the upper portion
of the cylinder. Since the ports are controlled by the piston, their opening and closing is
symmetric about bottom dead center. This type of scavenging has good performance at
low throttle and low engine speeds. The disadvantages are that the combustion chamber
becomes irregularly shaped with a high surface to volume ratio, increasing the susceptibility
for engine knock and possible overheating of the piston top.
With a cross-scavenging configuration, care must be taken to avoid short-circuiting.
At wide open throttle with increased intake pressure, the scavenging airflow has a tendency
to flow directly to the exhaust port. Notice in Figure 6.20 that without the deflector on
the piston top, the incoming air would have a tendency to simply go in and out of the
cylinder without displacing exhaust gas, that is, short circuit the intake and exhaust process.
Insertion of the deflector is intended to force the gas to turn and mix with the exhaust gas,
thus expelling a mixture of air and exhaust.
Experimental data suggest that the best scavenging that can be achieved via the cross
scavenging method occurs when there is perfect mixing in which the fresh air introduced
successively dilutes the residual exhaust gas. If sufficient air is used, at the end of scavenging
an acceptable scavenging efficiency is then achieved.
Loop scavenging was developed by Adolf Schnurle, a German engineer, in 1926. With
loop or Schnurle scavenging, the intake ports are angled and located 90--180◦ apart from
the exhaust ports. The top of the piston is relatively flat, reducing the piston overheating
issues. The angled intake ports produce a swirling scavenging flow that loops upward
and around the combustion chamber and then downward to the exhaust port. Again, the
ports are piston controlled. Numerical modeling of the fuel--air mixing in loop-scavenged
engines is discussed further in Kim et al. (2007).
Airflow in Two-Stroke Engines
187
Piston position
screw
Blower
Laminar flow
meter
Plenum
Cylinder
Thermocouple
Manometer for
flow meter
Manometer for port To atmosphere
pressure ratio
Figure 6.21 Flow bench measurement of effective flow areas and discharge coefficients of
piston-controlled ports.
Uniflow scavenging is a scavenging method that uses ports for the intake and valves
for the exhaust. The fresh air charge is admitted through piston-controlled ports near bottom
dead center and the exhaust gas exits through exhaust valves located in the cylinder head, so
the flow is ideally unidirectional. In theory, this method could result in perfect scavenging
in which the incoming air displaces the exhaust gas without any mixing occurring between
the incoming gas and the exhaust gas. The inlet ports are angled and located around the
entire periphery of the cylinder, producing a swirling flow in the cylinder. Also, since the
exhaust valves can be operated independently of the piston, it is possible to have greater
control of the compression ratio and compression pressure. Due to its greater mechanical
complexity compared with loop scavenging, uniflow scavenging is primarily used on large
marine diesel two-stroke engines. For current applications of uniflow scavenging, the reader
is referred to the numerical and experimental investigations of marine diesel engines in
Anderson et. al. (2013), and of vehicular diesel engines in Laget et al. (2013).
As inspection of Figure 6.20 reveals, more often than not, two-stroke engines use
piston-controlled ports rather than cam-actuated valves to admit the fresh charge and expel
the exhaust. Therefore, for two-stroke flow analysis and modeling, one must specify the
effective flow areas of the ports as functions of crank angle. A steady-flow apparatus for
determining the effective flow area of piston-controlled ports is shown in Figure 6.21.
Note the similarity with a valve flow bench apparatus. Solution of Equation 6.42 yields the
effective port area 𝐴f from measurements of the mass flow rate and the pressure ratio:
[
(( )
( )(𝛾+1)∕𝛾 )]1∕2
𝑃v
𝑃v 2∕𝛾
2
(6.42)
−
𝑚̇ = 𝜌o 𝐴f 𝑐o
𝛾 −1
𝑃o
𝑃o
Some measured discharge coefficients, using the exposed geometric port area as the
reference area, for a piston-controlled inlet port are shown in Figure 6.22. Part (b) of
Figure 6.22 shows an important difference between results obtained for poppet valves and
those obtained for simple ports. The discharge coefficient increases with Mach number,
whereas, with poppet valves, it is nearly independent of Mach number. As the Reynolds
number is not constant in Figure 6.22, the attribution of the observed effects to Mach
number tacitly assumes that there is no dependence upon Reynolds number.
Crankcase, inlet, and exhaust pressures are plotted in Figures 6.23 and 6.24 for a loopscavenged two-stroke motorcycle engine with piston-controlled induction operating at 𝑁 =
4000 rpm. Finite-amplitude pressure waves occur in the intake and exhaust pipes. The
Fuel and Airflow in the Cylinder
0.5 Y
Bore = 4 Y
Y
Port
form
C.L. cylinder
0.8
Cd 0.7
0.6 Y
0.6
0
0.2
0.4
0.6
0.8
1.0
Fraction open
(a)
0.9
Cd
Fraction open 1.00
0.6
0.9
Cd
0.60
0.6
0.9
Cd
0.20
0.6
0
0.78 1.05
1.36
2.15
Mach number
(b)
Figure 6.22 Port discharge coefficient. (a) Variation with port opening at low Mach number.
(b) Variation with Mach number based on velocity and sound speed at the throat. (Annand and Roe,
1974.)
TDC
IC
TO
BDC
TC
IO
TDC
1.50
Crankcase
1.00
0.75
1.25
Figure 6.23 Crankcase
and inlet pressure profiles
for a two-stroke motorcycle
engine. Adapted from Blair
and Ashe (1976).
0.50
1.00
Inlet
0.75
0
45
90
135
180
225
Crank angle
270
315
360
P/Po
1.25
P/Po
188
189
Airflow in Two-Stroke Engines
EO TO
TDC
BDC
TC EC
TDC
2.0
P/Po
2.5
1.5
Cylinder
1.0
Figure 6.24 Cylinder and
exhaust pressure profiles
for a two-stroke
motorcycle engine.
Adapted from Blair and
Ashe (1976).
P/Po
1.5
1.0
Exhaust
0.5
0
45
90
135 180 225
Crank angle
270
315
360
pressure waves are a significant influence in the performance of two-stroke engines, and
thus need to be considered in the design of the intake and exhaust manifolds. Figure 6.23
indicates that the crankcase pressure increases fairly linearly as the piston moves downward
until the transfer port is uncovered (TO), increases with an inlet pressure pulse, then
continues to decrease. Figure 6.24 shows a positive fluctuation in the cylinder pressure as
a plugging pulse returns in the exhaust. Additional information about two-stroke exhaust
tuning is given in Adair et al. (2006).
Performance Parameters
The following discussion provides performance terminology according to SAE recommended practice. There are two reference masses used in two-stroke scavenging analyses.
These are 𝑚o , the mass of delivered charge in an ideal scavenging process using the displacement volume 𝑉d and ambient air (or mixture) density, 𝜌𝑖 . This reference mass is useful
in experimental work.
𝑚o = 𝜌 i 𝑉 d
(6.43)
and 𝑚tr , the trapped, that is, actual mass of gas in the cylinder at a given instant, including
the delivered and the residual gas mass, useful in computational analysis. The relative
charge, 𝑅c , is defined as the ratio of these two masses:
𝑚
𝑅c = tr
(6.44)
𝑚o
The trapped air--fuel ratio is a measure used to characterize the state of the mixture at
the beginning of combustion:
mass of air retained
(6.45)
mass of f uel retained
The delivery ratio, 𝐷r , is the ratio of the actual mass of delivered charge to the ideal
mass of delivered charge. It has values ranging from zero at intake port opening to values
greater than one at exhaust port close depending on the amount of intake flow pressurization.
AF|tr =
𝐷r =
mass of delivered charge
𝜌i 𝑉 d
(6.46)
190
Fuel and Airflow in the Cylinder
The scavenging ratio, 𝑆r , is the ratio of the actual mass of delivered charge to the ideal
mass of delivered charge, using the entire cylinder volume and ambient air (or mixture)
density.
𝑆r =
mass of delivered charge
𝜌i 𝑉cyl
(6.47)
The trapping efficiency, 𝜂tr , is the fraction of the delivered air (or mixture) retained in
the cylinder at exhaust port close:
𝜂tr =
mass of delivered charge retained
mass of delivered charge
(6.48)
The scavenging efficiency, 𝜂sc , the ratio of the delivered charge retained to the mass
of gas in the cylinder, is a measure of the replacement of the burnt exhaust gas with fresh
charge at a given instant. It is used to compare the performance of various port and piston
geometries as a function of the delivery or scavenging ratio.
𝜂sc =
mass of delivered charge retained
𝑚tr
(6.49)
The charging efficiency, 𝜂ch , the ratio of the delivered charge retained to the ideal mass
of delivered charge, is a measure of the efficiency of the filling process:
𝜂ch =
mass of delivered charge retained
𝑚o
(6.50)
The purity is defined as the fraction of air in the trapped cylinder charge:
=
mass of air in trapped cylinder charge
𝑚tr
(6.51)
The parameters are defined for fuel-injected engines. For fuel-inducted engines, as
with a throttle-body injector, note that the air--fuel mixture is to be substituted for the air,
and the mixture density at ambient pressure and temperature is to be substituted for the
ambient air density.
The above parameters are not independent of each other. The scavenging efficiency
𝜂sc , charging efficiency 𝜂ch , and trapping efficiency 𝜂tr are all measures of the success
in clearing the cylinder of residual gases from the preceding cycle and as such can be
mathematically related. By definition it follows that the trapping efficiency 𝜂tr is
𝜂tr =
𝜂ch
𝜂 𝑅
= sc c
𝐷r
𝐷r
(6.52)
With excess air, the purity  and scavenging efficiency 𝜂sc differ because of the excess
air 𝜆 in the residual gas. It can be shown that (Schweitzer, 1949)
𝜂sc = 
𝜂sc =
if 𝜆 ≤ 1
1
1 + 𝜆(1∕ − 1)
(6.53)
if 𝜆 > 1
The scavenging efficiency is less than or equal to the purity. However, as the difference is
usually small, the two quantities are often confused.
Finally, the residual mass fraction 𝑓 required for thermodynamic analysis is
𝑓 = 1 − 𝜂sc
(6.54)
191
Airflow in Two-Stroke Engines
Trapping efficiency
1.0
A
C
B
0
1.0
r
r–1
Delivery ratio (Dr)
Scavenging efficiency
(a)
1.0
A: Perfect scavenging
B: Short circuiting
C: Perfect mixing
A
C
B
0
Figure 6.25 Two-stroke
scavenging and trapping
efficiencies.
1.0
r
r–1
Delivery ratio (Dr)
(b)
Two-Stroke Scavenging Models
In this section, we will use three simple algebraic mixing models corresponding to perfect
displacement, short-circuiting, and perfect mixing to find the relationships between the
scavenging efficiency and the delivery ratio. These models are approximations of the actual
scavenging process as they assume the process occurs at constant volume, temperature,
and pressure in the cylinder. Experimental measurements of scavenging efficiency as a
function of delivery ratio usually lie somewhere between the perfect displacement and
perfect mixing limiting cases.
Let us consider first the case of perfect scavenging. In this ideal case no mixing occurs,
and the inlet air simply displaces the exiting exhaust gas. The trapping and scavenging
efficiencies as functions of the delivery ratio are given in Figure 6.25.
At a delivery ratio 𝐷r given by
𝐷r =
𝑉bdc
𝑟
=
𝑉d
𝑟−1
(6.55)
the cylinder volume at bottom center is filled with pure air ( = 𝜂sc = 1.0) , and if any
more air is delivered, it is not retained. This occurs at a delivery ratio greater than one (see
curve A in Figure 6.25) and dependent upon the compression ratio because the delivery
ratio is defined in terms of the displacement volume 𝑉d rather than the maximum cylinder
192
Fuel and Airflow in the Cylinder
volume 𝑉cyl corresponding to bottom dead center. Therefore,
For 𝐷r ≤ 𝑟∕(𝑟 − 1),
𝜂sc = 𝐷r ∕𝑅c
𝜂tr = 1
For 𝐷r > 𝑟∕(𝑟 − 1),
𝜂sc = 1
(6.56)
𝜂tr = 𝑅c ∕𝐷r
In the case of short-circuiting, the air initially displaces all the gas within the path of
the short circuit and then simply flows into and out of the cylinder along that path. Thus,
initially, the scavenging efficiency 𝜂sc increases with delivery ratio as if scavenging were
perfect. The scavenging efficiency then remains constant once the path has been displaced,
see curve B in Figure 6.25.
For the case of perfect mixing, the first air to come in is assumed to be mixed with the
exhaust gasses to form a homogeneous mixture. The composition of the mixture leaving
the cylinder through the exhaust ports is the same as the instantaneous composition of the
in-cylinder mixture. Thus, the first gas expelled is nearly all residual gas. As the scavenging
process proceeds, the gas being expelled has an increasing concentration of fresh charge,
decreasing the trapping efficiency.
The scavenging and trapping efficiencies as a function of the delivery ratio can be
expressed via a mixing analysis based on the conservation of delivered air. Let 𝑚a denotes
delivered air, 𝑚′a denotes delivered air retained, and 𝑚 denotes the mass of the in-cylinder
mixture. The instantaneous mass fraction of delivered air retained is
𝑚′
(6.57)
𝑥= a
𝑚
and the airflow rate out of the cylinder is
𝑚̇ ′a, out = 𝑥 𝑚̇ out
(6.58)
The air mass continuity equation, Equation 6.59, is
𝑑𝑚′a
𝑑𝑡
= 𝑚̇ ′a, in − 𝑚̇ ′a, out
= 𝑚̇ in − 𝑥 𝑚̇ out
(6.59)
= 𝑚̇ in (1 − 𝑥)
The time derivative of Equation 6.57 is
𝑑𝑚′a
𝑑𝑚
𝑑𝑥
𝑑𝑥
=𝑥
+𝑚
=𝑚
(6.60)
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
assuming steady flow with equal mass flow rates into and out of the cylinder. Therefore,
𝑚
𝑑𝑥
= 𝑚̇ in (1 − 𝑥)
𝑑𝑡
(6.61)
separating variables,
𝑚̇
𝑑𝑥
= in 𝑑𝑡
1−𝑥
𝑚
Integrating over the scavenging event, where at exhaust port close (ec), 𝑥 = 𝜂sc
ec
ln(1 − 𝜂sc ) = −
∫eo
𝑚̇ in
𝑑𝑡
𝑚
(6.62)
(6.63)
References
193
Figure 6.26 Two-stroke
scavenging efficiency
versus engine speed.
Adapted from Blair and
Ashe (1976).
Scavenging efficiency
1.0
0.9
0.8
0.7
3000
4000
5000
6000
7000
Engine speed (rpm)
Since
ec
∫eo
𝐷
𝑚̇ in
𝑚
𝑑𝑡 = in = r ,
𝑚
𝑚tr
𝑅c
(6.64)
the scavenging and trapping efficiencies thus are
(
)
−𝐷r
𝜂sc = 1 − exp
𝑅c
[
(
)]
𝑅c
−𝐷r
𝜂tr =
1 − exp
𝐷r
𝑅c
(6.65)
The perfect mixing curves (C) are drawn in Figure 6.25 accordingly. The measured
and predicted scavenging efficiencies using the perfect mixing model are compared in
Figure 6.26 for a two-stroke motorcycle engine. The test engine is loop scavenged with
piston-controlled induction. The scavenging efficiencies are about 90%. A more detailed
review of scavenging modeling is given in Sher (1990).
State-of-the-art models use the differential mass and momentum conservation equations to predict the fluid flow and mixing conditions in a two-stroke engine as a function
of the engine speed, and port and cylinder geometry. For further information regarding
airflow in two-stroke engines, including the unsteady compressible flow characteristics of
two-stroke engines, the reader is referred to the books by Heywood and Sher (1999), and
Blair (1996).
6.8 REFERENCES
ADAIR, J., D. OLSEN, and A. KIRKPATRICK (2006), ‘‘Exhaust Tuning for Large Bore 2-Stroke Cycle
Natural Gas Engines,’’ Int. J. Engine Res., Vol. 7, Issue 2, pp. 131--141.
ADRIAN, R. (1991), ‘‘Particle-Imaging Techniques for Experimental Fluid Mechanics,’’ Ann. Rev.
Fluid Mech., Vol. 23, pp. 261--304.
ANDERSON, F., J. HULT, K. NOGENMYR, and S. MAYER (2013), ‘‘Numerical Investigation of the Scavenging Process in Marine Two-Stroke Diesel Engines,’’ SAE paper 2013-01-2647.
ANNAND, W. and G. ROE (1974), Gas Flow in the Internal Combustion Engine, G. T. Foulis, Somerset,
England.
BAUMGARTEN, C. (2006), Mixture Formation in Internal Combustion Engines, Springer-Verlag Publishing, Berlin.
BELAIRE, R., R. DAVIS, J. KENT, and R. TABACZYNSKI (1983), ‘‘Combustion Chamber Effects on Burn
Rates in a High Swirl Spark Ignition Engine,’’ SAE paper 830335.
194
Fuel and Airflow in the Cylinder
BLAIR, G. and M. ASHE (1976), ‘‘The Unsteady Gas Exchange Characteristics of a Two-Cycle
Engine,’’ SAE paper 760644.
BLAIR, G. P. (1996), Design and Simulation of Two Stroke Engines, SAE International, Warrendale,
Pennsylvania.
CONVERGE 2.1 MANUAL (2013), Convergent Science, Inc., Madison, Wisconsin.
DENT, J. C. (1971), ‘‘Basis for the Comparison of Various Experimental Methods for Studying Spray
Penetration,’’ SAE paper 710571.
FLUENT USERS MANUAL (2014), Fluent Incorporated, Hanover, New Hampshire.
GARNIER, E., N. ADAMS, and P. SAGAUT (2009), Large Eddy Simulation for Compressible Flows,
Springer-Verlag Publishing, Berlin.
HAN, Z. and R. REITZ (1995), ‘‘Turbulence Modeling of Internal Combustion Engines using RNG
𝑘 − 𝜖 Models,’’ Comb. Sci. Tech., Vol. 106, pp. 207--295.
HEYWOOD, J. and E. SHER (1999), The Two-Stroke Cycle Engine, SAE International, Warrendale,
Pennsylvania.
KAJIYAMA, K., K. NISHIDA, A. MURAKAMI, M. ARAI, and H. HIROYASU (1984), ‘‘An Analysis of Swirling
Flow in Cylinder for Predicting D. I. Diesel Engine Performance,’’ SAE paper 840518.
KAWASHIMA, J., H. OGAWA, and Y. TSURU (1998), ‘‘Research on a Variable Swirl Intake Port for
4-Valve High Speed DI Diesel Engines,’’ SAE paper 982680.
KIM, G., A. KIRKPATRICK, and C. MITCHELL, 2004, ‘‘Computational Modeling of Natural Gas Injection
in a Large-bore Engine,’’ ASME J. Eng. Gas Turbines Power, Vol. 126, No. 3, pp. 656--654.
KIM, G., A. KIRKPATRICK, and C. MITCHELL, 2007, ‘‘Supersonic Injection Virtual Valve Design for
Three Dimensional Numerical Simulation of a Large-bore Natural Gas Engine,’’ ASME J. Eng.
Gas Turbines Power, Vol. 129, No. 4, pp. 1065--1071.
LAGET, O., C. TERNEL, J. THIROT, and S. CHARMASSON (2013), ‘‘Preliminary Design of a Two-Stroke
Uniflow Diesel Engine for Passenger Car,’’ SAE Int. J. Engines, Vol. 6, No. 1, p. 596.
LIOU, T. M., M. HALL, D. A. SANTAVICCA, and F. N. BRACCO (1984), ‘‘Laser Doppler Velocimetry
Measurements in Valved and Ported Engines,’’ SAE paper 840375.
LI, Y., A. KIRKPATRICK, C. MITCHELL, and B. WILLSON (2004), ‘‘Characteristic and Computational
Fluid Dynamics Modeling of High Pressure Gas Jet Injection,’’ ASME J. Eng. Gas Turbines
Power, Vol. 126, No. 1, pp. 1--6.
LUMLEY, J. (1999), Engines: An Introduction, Cambridge University Press, Cambridge, England.
LUMLEY, J. (2001), ‘‘Early Work on Fluid Mechanics in the IC Engine,’’ Annual Rev. Fluid Mech,
Vol. 33, pp. 319--338.
OBERT, E. F. (1973), Internal Combustion Engines and Air Pollution, Harper & Row, New York, pp.
388--389.
OLSEN D. and A. KIRKPATRICK (2008), ‘‘Experimental Examination of Prechamber Heat Release in
a Large Bore Natural Gas Engine,’’ ASME J. Eng. Gas Turbines Power, Vol. 130, No. 5, pp.
052802: 1--7.
REITZ, R. and F. BRACCO (1979), ‘‘On the Dependence of Spray Angle and Other Spray Parameters
on Nozzle Design and Operating Conditions,’’ SAE Paper 790494.
SCHWEITZER, P. H. (1949), Scavenging of Two-Stroke Diesel Engines, Macmillan, New York.
SHER, E. (1990), ‘‘Scavenging the Two-Stroke Engine,’’ Prog. Energy Combust. Sci., Vol. 16, pp.
95--124.
SHI, Y., H. GE, and R. REITZ (2011), Computational Optimization of Internal Combustion Engines,
Springer-Verlag, London, England.
STAR-CD USERS MANUAL (2014), Computational Dynamics, Inc., London, England.
TAKAGI, Y., T. ITOH, S. MURANAKA, A. IIYAMA et al., (1998), ‘‘Simultaneous Attainment of Low Fuel
Consumption, High Output Power, and Low Exhaust Emissions in Direct Injection SI Engines,’’
SAE Paper 980149.
TAYLOR, C. (1985), The Internal Combustion Engine in Theory and Practice, Vols. 1 and 2, MIT
Press, Cambridge, Massachusetts.
TOWNSEND, A. (1976), The Structure of Turbulent Shear Flow, Cambridge University Press, Cambridge, England.
Homework
195
UZKAN, T., C. BORGNAKKE, and T. MOREL (1983), ‘‘Characterization of Flow Produced by a HighSwirl Inlet Port,’’ SAE paper 830266.
VECTIS USERS MANUAL (2014), Ricardo Software, Inc., Burr Ridge, Illinois.
WILCOX, D. (2006), Turbulence Modeling for CFD, Third Edition, DCW Industries, La Canada,
California.
6.9 HOMEWORK
6.1
A four-cylinder, four-stroke, 3.0 L port-injected spark ignition engine is running at 2200
rpm on a stoichiometric mix of octane and standard air at 100 kPa and 298 K. If the average
octane flowrate is 3.0 g/s, (a) what is the mass of fuel entering each cylinder per cycle?,
(b) what is the volumetric efficiency, and (c) assuming complete combustion, what is the
rate of heat release?
6.2
A carburetor has a pressure drop of 0.025 bar and a fuel--air ratio FA = 0.06 at a demand
𝐷c = 0.4. (a) What is the fuel--air effective area ratio 𝐴f ∕𝐴a ? (b) If the demand changes to
𝐷c = 0.6, what is the change in the fuel--air ratio FA?
6.3
Carburetor venturis are sized assuming the maximum quasi-steady flow during the intake
stroke is twice the average. Estimate the venturi throat diameter required for a four-cylinder
5.0 L engine with a volumetric efficiency of 0.9, and maximum speed of 6000 rpm. State
clearly the assumptions you need to make.
6.4
What is the injection duration (deg) needed for fuel injection in a single cylinder diesel
engine operating at 1500 rpm so that the engine produces 50 kW? The engine bsfc is 0.22
kg/kWh. Assume incompressible fuel flow, an average cylinder pressure of 50 bar during
the fuel injection, an injector effective area 𝐴f = 1.0 mm2 , and an injection pressure equal
to 550 bar.
6.5
If the diesel injector in problem 6.4 has a nozzle diameter 𝑑n of 0.30 mm and
length 𝐿n of 1.0 mm, what is the spray angle? Assume the cylinder temperature is
800 K.
6.6
For problem 6.5, plot the spray tip penetration versus time, and determine the time and
crank angle interval for the fuel to reach the cylinder wall for an engine with a bore of
150 mm operating at 1500 rpm.
6.7
A diesel fuel injector has a total orifice area of 0.15 mm2 with an average 𝐶d = 0.60.
a.) If the average pressure difference between the fuel injector and cylinder is 400 bar,
what is the amount of diesel fuel injected over a 8 ms period? b.) If the engine has a
50% thermal efficiency and operates at 1500 rpm, estimate the powered produced by the
engine.
6.8
To illustrate the effect of combustion chamber geometry on swirl amplification consider an
axisymmetric engine where at bottom center the velocity field of the air inside the cylinder
is approximately 𝑣r = 𝑣𝑧 = 0 and 𝑣𝜃 = 𝑉o (2𝑟∕𝑏). The cylinder has a bore 𝑏, and the piston
has a disk-shaped bowl of diameter 𝑑 and depth ℎ. The motion is said to be solid body
since the gas is swirling as though it were a solid. If at top dead center the motion is also
solid body and angular momentum is conserved during compression, what is the ratio of
the initial to final swirl speed, 𝜔bdc ∕𝜔tdc , as a function of the compression ratio and the
cylinder geometry? The moment of inertia of solid body rotation of a disk of diameter, 𝑑,
and depth, ℎ, is 𝐼 = 𝜋𝜌ℎ𝑑 4 ∕32.
196
Fuel and Airflow in the Cylinder
6.9
An engine has a mean piston speed 𝑈̄ p of 10.0 m/s and a clearance volume height ℎ of 5 mm.
What is the characteristic length 𝐿, integral scale 𝑙, Taylor microscale 𝜆, and Kolmogorov
microscale 𝜂 at the end of compression? Compare your calculations with Example 6.2 and
explain the differences. Assume the fluid kinematic viscosity at the end of compression is
100 × 10−7 m2 /s and 𝐶𝜂 = 𝐶𝜆 = 1, 𝐶l = 0.2.
6.10
A single-cylinder, two-stroke carbureted engine of 85 mm bore and 110 cm stroke is
operating at 2500 rpm. It has a compression ratio 𝑟 = 8, is fueled with gasoline, and is
running rich with an equivalence ratio 𝜙 = 1.2. If its indicated power is 20 kW with
inlet air temperature of 345 K, inlet pressure of 101 kPa and exhaust pressure of 105
kPa, compute its scavenging ratio 𝑆r and scavenging efficiency 𝜂sc . Use Figure (4.4) to
estimate the indicated thermal efficiency 𝜂Otto of an equivalent fuel--air cycle, and assume
𝜂∕𝜂Otto = 0.80.
Chapter
7
Combustion Processes
in Engines
7.1 INTRODUCTION
In this chapter, we examine combustion processes in spark ignition and compression
ignition engines. The combustion processes that occur in each of these types of engines
are very different. A spark ignition engine has a relatively homogeneous turbulent mixture
of fuel and air, which once ignited by a spark, sustains a reaction process that propagates
a flame in the form of a thin wrinkled sheet through the mixture. During the combustion
process, the energy release rate starts relatively slowly, increases to a maximum value,
and then decreases near the end of the combustion process. Accordingly, the cylinder
pressure increases to a maximum value after top dead center as the flame propagates across
the cylinder, and then decreases during the expansion stroke. The performance of spark
ignition engines is limited by the occurrence of an autoignition process called knock, which
constrains the maximum compression ratio and thus the overall engine efficiency.
On the other hand, a compression ignition engine has separate fuel and air streams
that combust as they are mixed together at a temperature greater than the autoignition
temperature. The combustion reaction, which produces a diffusion flame, takes place
at the interface between the fuel and the air. The energy release begins at a relatively
high value, and then decreases as the available oxygen is depleted. The performance of
compression ignition engines is limited by emissions of unburned hydrocarbons, including
soot.
The combustion processes in both spark and compression ignition engines are very
complex and depend on the type of fuel and the amount of air used in the combustion process. For example, the reaction pathways for the oxidation of a hydrocarbon fuel such as
paraffin, Cn H2n+2 , a major component of gasoline, can include at least 10,000 different reactions. To keep such reactions computationally tractable, the large detailed reaction mechanisms are reduced to mechanisms with less species and reaction numbers, as discussed by
Law et al. (2000).
Hydrocarbon reactions are generally grouped into three distinct steps. For example, the
first step in the combustion of a fuel molecule is breaking up carbon--carbon bonds in the
fuel molecule, forming alkenes (hydrocarbons with double carbon bonds) and hydrogen.
The second step is further oxidation to form CO and hydrogen. The third and last step is
the oxidation of CO to form CO2 . Most of the energy release occurs during the last step, a
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
197
198
Combustion Processes in Engines
step independent of the molecular mass of the fuel. Consequently, hydrocarbon paraffins
of different molecular mass have very similar heats of combustion.
Recently, with advances in injection and engine control technology, a variety of techniques, categorized as ‘‘low-temperature combustion’’ (LTC), have been developed to
combine the best aspects of spark and diesel combustion, that is, combining the nearly
homogeneous fuel--air mixture of a spark ignition engine with the higher compression of
a diesel engine to achieve higher engine efficiencies with lower emissions. For the most
part, these techniques are still in the research stage, but are very promising approaches.
7.2 COMBUSTION IN SPARK IGNITION ENGINES
Ignition
An energy source is needed to ignite the fuel--air mixture in a spark ignition engine.
Early low compression engines used flames and spark ignition for this purpose, and as the
compression ratio was increased, techniques such as high-voltage spark ignition, plasma
jets, and laser ignition have been adopted. Spark ignition is by far the dominant ignition technology used currently in internal combustion engines. J. Lenoir used spark plug
ignition to initiate combustion in his internal combustion engine in 1860, and R. Bosch
and N. Tesla independently developed high-voltage spark plug technology in 1898. In
1911, C. F. Kettering, an American engineer, developed the electrical starter motor and
high-voltage ignition for automobile engines, replacing the cumbersome hand cranked
magneto.
With spark ignition, combustion is initiated by an electrical discharge across an electrode gap. The spark discharge requires about 20,000--30,000 V, and uses a spark plug
configuration consisting of two electrodes, one grounded to the engine, and a center
electrode insulated with porcelain. The spark discharge process has four main phases:
predischarge, breakdown, arc, and glow. In the predischarge phase, energy is added to
the gas molecules to ionize them. The voltage increases to the point that current can flow
across the gap. The point at which current begins to flow through the ionized gas signals the onset of the breakdown phase. In the breakdown phase, the current increases to
about 102 A in a few nanoseconds, establishing a low-impedance pathway between the
electrodes. The sustainability of the plasma kernel is very dependent on the characteristics of the breakdown phase, since this phase has the highest voltage and current. In the
low-impedance arc phase, the voltage is reduced to about 100 V, and the current decreases
to about 1 A. The final phase of the spark discharge process is the glow discharge, which
lasts much longer than the previous phases, with a voltage of about 500 V and current of
about 0.1 A.
During the discharge process, electrical energy is transferred to the fuel--air mixture,
raising its temperature high enough to initiate local chemical reactions. After approximately
20--100 ms after onset of ionization, if the chemical energy release exceeds the conduction
heat transfer to the surrounding unburned mixture, the chemical reactions become selfsustaining, a glowing spherical flame kernel is formed, and a flame propagates radially
away from the spark plug. The amount of ignition energy required decreases with increasing
mixture temperature and pressure and is minimum at stoichiometric conditions. As the
mixture becomes more lean, the laminar flame speed decreases, increasing the conduction
heat transfer from the flame to the surrounding gas, and consequently increasing the
ignition energy required. Variations in the mixture composition between the electrodes will
contribute to variations in the flame propagation process, known as cyclic variation. Once
established, the flame can also be transported by bulk flow motion.
Combustion in Spark Ignition Engines
199
Combustion Visualization
A wide variety of imaging techniques have been used to visualize the combustion processes
in a spark ignition engine. In the late 1930s, Rassweiler and Withrow (1938) modified an
L-head cylinder of a spark ignition engine so that a quartz window could be installed
allowing an unobstructed view of the entire combustion space. Using high-speed motion
photography, they were able to record the combustion process in a homogeneous charge,
spark ignition engine. For the combustion process they photographed, ignition occured at
𝜃s = −25◦ , and no flame was visible until 𝜃s = −16◦ , 9◦ later. That 9◦ period was called the
ignition delay, Δ𝜃id . Once formed, the flame spread like a spherical wave into the unburned
gas with a ragged surface because of turbulence. The end of combustion at 𝜃s = +25◦ was
determined from simultaneous measurement of cylinder pressure.
More recently, Witze and Vilchis (1981) used a laser shadowgraph technique for
combustion visualization. Shadowgraph photography is a method of flow visualization
that shows contrasts due to differences in density of the flow. It does not record light
emitted by the flame, rather it records light transmitted through and refracted by the
gases. Figure 7.1 shows laser shadowgraph sequences for lean (𝜙 = 0.55) and slightly rich
(𝜙 = 1.1) combustion and Figure 7.2 presents the corresponding pressure profiles. A ragged
edge wave is seen propagating into the unburned mixture. Ignition delay (in degrees of
crank angle) is on the order of 10◦ for the rich case and 20◦ for the lean case. At 20◦ and 25◦
after ignition in the rich case, the width of the flame front is clearly discernible. The width
is more difficult to discern in the lean case because it is two or three times thicker. Thus,
a completely burned region does not appear until approximately 40◦ after ignition. At this
time, the whitest region is burned gas, the grayish region in front of the flame is unburned
gas, and the highly convoluted dark and white region is a mixture of burned, burning, and
unburned gas. There are also cycle-to-cycle variations in the flame propagation caused by
the random features of the flow field. For further reading, a recent review of various optical
diagnostic techniques is given in Soid and Zainal (2011).
Combustion Process Analysis
In a spark ignition engine, a flame propagates across a homogeneous fuel--air mixture. As
shown in Figure 7.3, the concentration of reactants decreases, and the temperature of the
mixture increases across the flame front. The flame front has two zones: a preheat zone
in front of the flame in which the temperature of the reactants is raised to the ignition
temperature by conduction heat transfer from the flame front into the unburned region, and
a narrow reaction zone that contains the flame front where the combustion takes place.
The energy release is negligible in the preheat zone. As the reactant’s temperature rises
in the reaction zone, the chemical reactions, which depend exponentially on temperature,
increase until the reactants are consumed and their concentration then decreases to zero,
forming the downstream side of the flame front.
There are a number of models of the homogeneous mixture combustion process in
spark ignition engines. The models include zero-dimensional models, for example, two
zone thermodynamic models that divide the combustion chamber into burned and unburned zones; quasi-dimensional models, for example, models that incorporate a turbulent
flame speed to determine rate of mass burned; and multidimensional models, for example,
flamelet, eddy break up, probability density function, and coherent flame models, which include combustion chamber geometry and associated boundary conditions. There are similar
models for compression ignition engines.
200
Combustion Processes in Engines
Figure 7.1 Laser shadowgraph of lean
(left) and rich (right) combustion (Witze
and Vilchis, 1981). Reprinted with
permission, SAE.
Combustion modeling and incorporation into CFD models is a very active area of
research. The CFD codes such as FLUENT, STAR-CD, and VECTIS that are referenced in
the previous chapter also contain combustion models. The computer program CHEMKIN
(Kee et al., 2006) is widely used in conjunction with CFD codes for combustion analysis
in internal combustion engines, as it is able to model a variety and number of reactions.
Combustion in Spark Ignition Engines
201
Figure 7.2 Pressure profiles for
Figure 7.1 (Witze and Vilchis,
1981). Reprinted with permission,
SAE.
Figure 7.3 Temperature and species
concentration profiles in laminar flames.
Adapted from Borman and Ragland (1999).
The combustion parameters incorporated into these models include the laminar flame
speed 𝑠l , flame thickness 𝛿l , the turbulent flame speed 𝑠t , and the turbulence intensity 𝑢t .
The laminar flame speed 𝑠l , or burning speed, is a well-defined characteristic of a fuel--air
mixture, and represents the speed at which a one-dimensional laminar flame propagates
into the unburned gas under nonturbulent and adiabatic conditions.
The laminar flame speed depends on the pressure, temperature, equivalence ratio,
and composition of the unburned gas. From measurements, Metghalchi and Keck (1982)
developed the following correlation, Equation 7.1, for the laminar flame speed, based on the
unburned (u) fuel--air properties, and valid for 𝑇u > 350 K. The fuel--air mixtures measured
were methanol, propane, isooctane, and indolene.
𝑠l = 𝑠l,o
(
𝑇u
𝑇u,o
)𝑎 (
𝑃
𝑃o
)𝑏
(1 − 2.1 𝑓 )
(7.1)
where 𝑇u,o = 298 K and 𝑃o = 1 atm, and 𝑓 is the residual fraction, accounting for residual
or recycled exhaust gases. The laminar flame speed was found to decrease linearly with
residual fraction. The reference flame speed 𝑠l,o (cm/s) is given by Equation 7.2:
𝑠l,o = 𝑐1 + 𝑐2 (𝜙 − 𝜙o )2
where 𝑐1 , 𝑐2 , and 𝜙o are given in Table 7.1.
(7.2)
Combustion Processes in Engines
Table 7.1 Curve-Fit Parameters for Laminar Flame Speed Correlation, Equation 7.2
Fuel
Methanol
Propane
Isooctane
𝑐1
𝑐2
𝜙o
36.92
34.22
26.32
−140.51
−138.65
−84.72
1.11
1.08
1.13
40
Isooctane
Methanol
35
Flame speed (cm/s)
202
30
25
20
Figure 7.4 Laminar flame
speed versus equivalence ratio
(Equation 7.1).
15
0.7
0.8
0.9
1
1.1
1.2
1.3
Fuel−air equivalence ratio
1.4
The parameters 𝑎 and 𝑏 depend on the equivalence ratio, and for the fuels tested by
Metghalchi and Keck are given by
𝑎 = 2.18 − 0.8(𝜙 − 1)
(7.3)
𝑏 = −0.16 + 0.22(𝜙 − 1)
(7.4)
Some of these dependencies are illustrated in Figures 7.4 and 7.5, which plots the above
correlation. The laminar flame speed that shows a maximum for slightly rich mixtures is a
strong function of unburned gas temperature 𝑇u and is a weak function of pressure 𝑃 . The
strong dependence of the laminar flame speed on unburned gas temperature is due to the
exponential relation between the reaction kinetics and temperature.
There are three regimes for turbulent flames. The regimes are wrinkled laminar flame,
flamelets in eddies, and distributed reaction. The characteristics of the regimes are outlined
in Table 7.2. Internal combustion engines operate in the wrinkled laminar flame and the
flamelets in eddies regimes, depending on the engine speed (Abraham et al., 1985).
In the wrinkled laminar flame regime, the flame thickness 𝛿l is thinner than the smallest
Kolmogorov microscale 𝜂, and the turbulent intensity 𝑢t is of the same order as the laminar
Table 7.2 Turbulent Flame Regimes
Wrinkled laminar flame
Flamelets in eddies
Distributed reactions
𝛿l < 𝜂
𝜂 < 𝛿l < 𝑙
𝛿l > 𝑙
𝑢t ∼ 𝑠l
𝑢t >> 𝑠l
Combustion in Spark Ignition Engines
140
203
P = 1 atm
P = 10 atm
Flame speed (cm/s)
120
100
80
60
40
Figure 7.5 Laminar flame speed
versus gas temperature
(Equation 7.1).
20
300
400
500
600
Unburned gas temperature (K)
700
flame speed 𝑠l . The effect of turbulence in the cylinder therefore is to wrinkle and distort the
laminar flame front. In the flow field, the turbulent vortices spread ignition sites via a ragged
edge wave emerging from the spark plug. The position of the flame front moves irregularly,
making the time average flame profile appear relatively thick, forming a ‘‘turbulent flame
brush.’’ For the turbulent flow conditions of Example 7.3, the scale of the wrinkles is about
1 mm, and the flame is less than 0.01 mm thick.
The turbulent flame speed can be from 3 to 30 times greater than the laminar flame
speed. In the wrinkled laminar flame regime, if one assumes that the area of the wrinkles
is proportional to the turbulence intensity, then the turbulent flame speed 𝑠t is simply the
sum of the laminar flame speed and the turbulence intensity, represented by Equation 7.5
(Damkohler, 1947).
𝑠t = 𝑠l + 𝑢t
(7.5)
A convenient way to conceptualize the flame propagation in the wrinkled laminar
flame regime is in terms of ink rollers. The ink roller model is shown in Figure 7.6. Imagine
a bunch of cylindrical rolls as depicted to represent eddies of a similar diameter in the
turbulent flow field. Now, consider ignition as being analogous to continuously depositing
a stream of ink at the periphery of one roll. The rollers are rotating, and as a result, the ink
spreads. A ragged edge wave emerges from the initial deposition site. The speed of the
propagation is proportional to the velocity at the edge of the vortices. The front will take
on a thickness determined by the speed of the rollers, their size, and the rate at which ink
seeps into the rolls. In the flow field, the flame thickness will depend on the vorticity, the
eddy sizes, and the laminar flame-spreading rate. As turbulence is a three-dimensional fluid
phenomena, the eddies in the flow field are more likely to resemble a mesh of spaghetti
than perfectly aligned ink rollers.
As discussed earlier, the turbulence intensity is proportional to the engine speed, so at
higher engine speeds, the turbulent flame region can transition from a wrinkled sheet to the
flamelets-in-eddies regime. In the flamelets-in-eddies regime, the flame thickness is greater
than the small eddy thickness 𝜂, but less than the integral thickness 𝑙. The turbulent intensity
is much greater than the laminar flame speed. The increased wrinkling can result in the
creation of pockets of unburned gas mixture. Accordingly, in this regime, the burning rate
is controlled by the turbulent mixing rate, that is, the integral length scale, not the chemical
reaction rate.
204
Combustion Processes in Engines
Figure 7.6 Ink roller model
of turbulent combustion.
The combustion also depends on the combustion chamber geometry. To illustrate the
effect of combustion chamber geometry, consider two limiting cases of combustion: (1)
in a sphere centrally ignited and (2) in a tube ignited at one end. Assume that the sphere
and the tube have the same volume. In each case, the flame will propagate as a ragged
spherical front of radius 𝑟f from the spark plug. In the sphere, the area of the front grows
as 𝑟2f . Thus, the entrainment rate gets faster and faster as the flame grows. On the other
hand, in the tube, the flame front will initially grow as 𝑟2f , but it will soon hit the walls and
be constrained to be more or less constant from then on. Thus, combustion in a sphere can
be expected to burn faster, that is, it will take less time to burn the charge. The maximum
cylinder pressure occurs at about the time that the flame reaches the cylinder wall. This is
also the point of largest flame surface area, with the maximum flow of unburned gases into
the flame.
Energy Release Analysis
The differential energy equation analysis introduced in Chapter 2 can be used to compute
the mass fraction burned 𝑥b , and the combustion duration 𝜃d if the cylinder pressure versus
crank angle is known. If one assumes thermal equilibrium at each crank angle, a uniform
mixture, ideal gas behavior, and cylinder wall heat transfer loss 𝑄w , the first law for a
single zone is
𝑑𝑄w
𝛾 𝑃 𝑑𝑉
𝑑𝑄
1 𝑉 𝑑𝑃
=
+
+
𝑑𝜃
𝛾 − 1 𝑑𝜃
𝛾 − 1 𝑑𝜃
𝑑𝜃
(7.6)
Equation 7.6 can be solved numerically to obtain the net energy release per unit crank angle
𝑑𝑄 / 𝑑𝜃. The mass fraction burned 𝑥b (𝜃) at any crank angle 𝜃 is then found from calculating
Combustion in Spark Ignition Engines
205
the integral of the energy release normalized by the total energy release integral:
𝑥b =
𝜃 𝑑𝑄
s 𝑑𝜃
𝜃e 𝑑𝑄
∫𝜃 𝑑𝜃
s
∫𝜃
𝑑𝜃
(7.7)
𝑑𝜃
Engine simulation programs typically use a energy release profile that has been curve-fitted
with a Wiebe function, Equation 7.8:
[ (
) ]
𝜃 − 𝜃s 𝑛
𝑥b (𝜃) = 1 − exp −𝑎
𝜃d
(7.8)
This single zone analysis can be extended to two zones by assuming that the combusting
mixture can be split into a burned and an unburned zone. The unburned zone includes the
gas mixture ahead of the flame and unburned gas within the flame. The burned zone includes
gas behind the flame and burned gas within the flame. Thus, the highly convoluted flame
structure observed via flow visualization is accounted for, and the analysis is limited in
principle only by the assumption that the mass of gas actually reacting is small.
In practice, the analysis is limited further by imprecise estimates of the heat transfer
and mass loss as well as experimental error in the pressure measurement. For example,
a model (Tabaczynski et al., 1980) has been developed to predict mass-fraction burned
curves from fundamental quantities such as the laminar flame speed of the fuel and the
turbulence intensity of the flow. Key to the analysis are the ‘‘ink roller’’ assumptions
that ignition sites are spread by turbulence and the laminar burnup of material between
shear layers occurs. Representative results are given in Figure 7.7, which show the ignition
delay and combustion duration portions of the mass-fraction burned curve increasing as
the equivalence ratio is decreased or the EGR increased.
Two aspects of the mass-fraction burned curve that are used to characterize the combustion are the ignition delay and the combustion duration. Figure 7.8 is a representative
plot of the ignition delay angle versus equivalence ratio. The ignition delay is defined in
this case as the crank angle change from spark firing to 1% mass fraction burned. The
ignition delay depends on spark timing, residual fraction, and equivalence ratio.
Consistent with observations made via flame photography, ignition delay increases
as the mixture is leaned out from stoichiometric. The ignition delay increases with spark
advance because the laminar flame speed decreases as a result of lower temperatures at
the time of spark, but it is not the sole effect, for the turbulent field is also different.
Likewise, the ignition delay increases as the mixture is diluted either by leaning the charge
or recirculating the exhaust. The change is proportionately less than the change in laminar
flame speed. This is due to the influence of combustion on the turbulence field as the flame
grows.
Figure 7.9 is a representative plot of combustion duration versus equivalence ratio.
The combustion duration in this case is defined as the crank angle change from 1 to 90%
burned fraction. Like ignition delay, the combustion duration depends on the equivalence
ratio, the residual fraction, and the spark timing.
The combustion duration also depends on the laminar flame speed, the turbulence
intensity of the flow, and the combustion chamber geometry. Minimizing the combustion
duration in an engine requires a high turbulence intensity (which is often achieved at
the expense of volumetric efficiency), a flame area that increases with distance from the
spark plug, and a centrally located spark plug to minimize flame travel. As one expects,
minimizing the combustion duration maximizes the work done, since the combustion
approaches constant volume, and it also lowers the octane level required. Figure 7.10
206
Combustion Processes in Engines
1.0
0.9
1.0
O=
0.9
Mass fraction burned
0.8
0.7
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–40
–20
0
20
40
Crank angle (deg)
60
80
60
80
(a)
1.0
0.9
Mass fraction burned
0.8
EGR (%) = 0
0.7
15
30
0.6
0.5
0.4
0.3
0.2
0.1
Figure 7.7 Representative
mass-fraction burned curves: (𝑎) for
varying equivalence ratio and (𝑏) for
varying EGR.
0
–40
–20
0
20
40
Crank angle (deg)
(b)
shows experimental results for three different combustion chamber shapes, each with a
compression ratio 𝑟 = 9, engine speed 𝑁 = 1000 rpm, and ignition at maximum torque.
Note that the bowl shaped combustion chamber had the shortest combustion duration and
the lowest required octane level.
7.3 ABNORMAL COMBUSTION (KNOCK) IN SPARK IGNITION ENGINES
Knock is the term used to describe an abnormal internal combustion engine phenomenon
that produces an audible high frequency pinging or ‘‘knocking’’ noise. The performance
of spark ignition engines is limited by the onset of knock. During knock, the unburned or
end gas auto ignites and combusts before the arrival of the flame front. This combustion
results from compression of the end gas by the expansion of the burned part of the charge,
raising the end gas temperature to the autoignition point.
Abnormal Combustion (Knock) in Spark Ignition Engines
Delivery ratio
0.8
0.5
0 s = –40
1600 rpm
30
0 s = –40
1250 rpm
0.40
0.40
20
0.30
0.20
0.10
10
0.30
0.20
0.10
0.40
0.30
0.20
0.10
Residual fraction
0s = –20
1600 rpm
Residual fraction
40
Residual fraction
Ignition delay (CA deg)
50
207
0
0.8
0.9
1.0
0.8
1.0
0.9
Equivalence ratio
0.8
0.9
1.0
Figure 7.8 Ignition delay versus equivalence ratio and residual fraction. Adapted from Young
(1980).
Delivery ratio
0.8
0.5
50
Os = –40
1600 rpm
Os = –40
1250 rpm
0.40
30
0.20
20
0.40
0.30
0.20
0.10
0.30
0.20
Residual fraction
0.30
0.40
Residual fraction
40
Residual fraction
Combustion duration (CA deg)
Os = –20
1600 rpm
0.10
10
0.10
0
0.8
0.9
1.0
0.8
0.9
1.0
Equivalence ratio
0.8
0.9
1.0
Figure 7.9 Combustion duration versus equivalence ratio and residual fraction. Adapted from
Young (1980).
The autoignition in the end gas creates pressure waves that travel through the combustion gases, producing a rapid pressure rise and extremely high localized temperatures.
The pressure waves can be of several different types. For example, they can take the
form of finite amplitude supersonic pressure waves that decay rapidly to smaller amplitude
Combustion Processes in Engines
Pressure (bar)
208
50
40
a
30
Figure 7.10 Effect of
combustion chamber
geometry on combustion
duration and octane
requirement. Adapted
from Caris et al. (1956).
b
20
Ignition
c
a: Disk (95 ON)
b: Wedge (89 ON)
a
–20
c
b
–10
10
c: Bowl (73 ON)
0
10
20
30
Crank angle (deg)
resonant sound waves. The attendant rapid fluctuations in pressure can be a serious problem, as they can disrupt the cylinder thermal boundary layers causing higher piston surface
temperatures, resulting in surface erosion and failure. The onset of knock puts a constraint
on spark ignition engine performance, since it limits the maximum compression ratio and
thus the engine power.
Knock occurrence has been found to be dependent on many variables, including
engine speed, fuel properties, combustion chamber design, equivalence ratio, and intake
air temperature and pressure. The most important parameter is the end gas temperature, as
the tendency to knock is directly proportional to the end gas temperature.
Characteristic cylinder pressure profiles for normal and knocking combustion are
shown in Figure 7.11. The knock spectra of the pressure profiles depend on the cylinder
chamber geometry and the speed of sound in the cylinder gases. Classic measurements by
Draper (1938) indicate that the dominant acoustic frequency is the first tangential mode
(1T) of vibration, followed by higher order harmonics. The 1T acoustic mode is one in
which there are pressure fluctuations both in the tangential and radial directions. For a
cylindrical combustion chamber, the dominant acoustic frequency 𝑓 (Hz) is given by the
following equation:
𝑓=
𝑐𝑛
𝜋𝑏
(7.9)
where 𝑐 is the speed of sound (m/s), 𝑛 is the wave mode eigenvalue, equal to 1.841 for the
1T mode, and 𝑏 is the cylinder diameter. Typical knock frequency spectra are in the 2--10
kHz range for automobile size engines.
Using a single-cylinder research engine, the unburned end gas in a high swirl, homogeneous charge engine has been isolated in the center of the combustion chamber by
simultaneous ignition at four equally spaced spark plugs mounted in the cylinder wall.
High-speed schlieren photographs reveal that under knocking conditions, the flame spread
occurs much faster than normal. Figure 7.12 shows the dramatic change in the schlieren
pattern just before and just after ignition. The top line in the figure is the pressure versus
crank angle, starting at TDC. It took about 2 ms for the flames to spread from the spark
plugs to the position shown in the leftmost photograph just before knock, whereas it took
only 0.1 ms to propagate through the end gas once autoignition occurred, as shown in the
rightmost photograph. In this case, neither shock nor detonation waves were observed.
Abnormal Combustion (Knock) in Spark Ignition Engines
209
Pressure
Intense knock
Slight knock
Normal
combustion
Figure 7.11 Pressure profiles for knocking
conditions. Adapted from Douaud and Eyzat
(1977).
–20
tdc
20
40
60
Crank angle
Figure 7.12 Schlieren
photographs of knock
process (Smith et al., 1984).
This is because the unburned gas involved is at an elevated temperature, so the laminar
flame speed is substantially increased. More importantly, however, several autoignition
sites appear almost simultaneously.
In these experiments, temperature measurements have been made of the end gas
using a laser-based technique. For temperatures less than 1100 K, coherent antistokes
Raman spectroscopy (CARS) is used, and at the higher temperatures, spontaneous Raman
scattering is used. The results, shown in Figure 7.13, show that the end gas temperature,
like the pressure, undergoes an abrupt change in the rate of change at the knock point.
They also show that the temperature continues to rise even after the 0.1 ms required for
the homogeneous ignition sites and the flame propagation to have consumed the end gas.
Clearly, oxidation is not complete in the after-knock photograph of Figure 7.12.
Rapid compression machines (RCM) have been used to study combustion processes.
The RCM has been used for many years to study autoignition phenomena and simulate the
compression stroke of an internal combustion engine. With an RCM, a reactive mixture
is introduced into a chamber, rapidly compressed by a moving piston, and the resulting
210
Combustion Processes in Engines
2600
Temperature (K)
2200
1800
1400
Knock point
1000
Figure 7.13 Temperature history of the end gas in
Figure 7.12 as determined by CARS and Raman
scattering (Smith et al., 1984).
0
20
40
60
80
100
% Mass fraction burned
temperature and pressure profiles measured. As discussed in Lee and Hochgreb (1998),
the experimental results are used to verify proposed chemical kinetic mechanisms for
autoignition.
Knock Modeling
One way to model engine knock is to suppose that there exists a critical mass fraction of
combustion precursors that if attained anywhere within the end gas, that is, the unburned
portion of the fuel--air mixture, will cause autoignition (Downs, 1951). Knock will then
occur prior to the end of normal combustion if the integrated rate of formation equals this
critical mass fraction.
The normalized rate of formation of precursors is represented by an equation, Equation
7.10 of the following form, where 𝑥p is the mass fraction of precursors, 𝑥c is the critical
mass fraction, and 𝑇u is the temperature of the unburned fuel--air mixture:
(
)
−𝐵
1 𝑑𝑥p
𝑛
= 𝐴 𝑃 exp
(7.10)
𝑥c 𝑑𝑡
𝑇u
The empirical constants 𝐴, 𝐵, and 𝑛 are determined from a set of experimental results.
Values of 𝐴 = 50.5, 𝐵 = 3800, and 𝑛 = 1.7 were obtained by Douaud and Eyzat (1978).
Like constants in algebraic burning laws, these constants will vary from engine to engine
and from fuel to fuel.
We can define the extent of precursor reaction 𝜁 as the ratio of the precursor mass
fraction to the critical mass fraction:
𝜁 = 𝑥p ∕𝑥c
(7.11)
so upon differentiation with respect to time 𝑡,
𝑑𝜁
1 𝑑𝑥p
=
𝑑𝑡
𝑥c 𝑑𝑡
(7.12)
Abnormal Combustion (Knock) in Spark Ignition Engines
211
and
𝜁=
∫0
𝑡
𝑡 𝑑𝑥
𝑑𝜁
p
1
𝑑𝑡 =
𝑑𝑡
𝑑𝑡
𝑥c ∫0 𝑑𝑡
(7.13)
The extent of precursor reaction 𝜁 can be expressed in terms of crank angle 𝜃 and
engine speed 𝜃.̇ Since 𝜃̇ = 𝑑𝜃∕𝑑t,
𝜁=
𝜃
𝑑𝜁
1
𝑑𝜃
̇𝜃 ∫𝜃 𝑑𝑡
s
(7.14)
If at any time prior to the end of combustion 𝜁 reaches 1, knock is said to occur and
the remaining unburned end gas burns instantaneously. The minimum engine speed 𝜃̇ min ,
above which there is not sufficient precursor formation time for knock to occur, is thus
𝜃
𝜃̇ min =
𝑑𝜁
𝑑𝜃
∫𝜃s 𝑑𝑡
(7.15)
The reaction rate is an extremely strong function of the temperature (See homework
problem 7.5). Indeed, at temperatures characteristic of the intake manifold, the rate of
formation of precursors is negligible. Since the combustion in automotive class engines
occurs over times of order 10−2 s, not until the rates approach 100 s−1 will knock occur
with isooctane.
EXAMPLE 7.1
Spark Ignition Engine Knock
For a constant pressure combustion at 10 bar and unburned gas temperature of 1100 K,
what is the precursor formation rate 𝑑𝜁 ∕𝑑𝑡 and the minimum engine speed 𝜃̇ min if the
combustion duration is 40◦ ? Assume 𝐴 = 50.5, 𝐵 = 3800, and 𝑛 = 1.7.
SOLUTION
𝑑𝜁
= 𝐴 𝑃 𝑛 exp
𝑑𝑡
(
−𝐵
𝑇u
)
)
(
3800
= (50.5)(10)1.7 exp −
1100
= 80.0 s−1
𝜃
𝜃̇ min =
=
𝑑𝜁
𝑑𝜃
∫𝜃s 𝑑𝑡
𝑑𝜁
(𝜃 − 𝜃s )
𝑑𝑡
= (80.0)(40)
= 3200 deg∕s = 533 rpm
212
Combustion Processes in Engines
A reduced kinetic model of autoignition chemistry and knock, known as the Shell
model, frequently used in CFD simulations, is given by Halstead et al. (1977), and compared
with a detailed kinetic knock model by Cowart et al. (1990).
Octane Number
To provide a standard measure of a fuel’s knock characteristics, a scale has been devised
in which fuels are assigned an octane number, ON. The first knock scale was developed
by Ricardo (Ricardo, 1921), and fuels were rated by the highest useful compression ratio
(HUCR) that could be used in an engine under a given set of conditions. Since the HUCR
depended on the given engine, it was supplemented by a knock scale developed in the
1920s by the American Society for Testing Materials (ASTM) Cooperative Fuel Research
Committee (CFR).
The Cooperative Fuel Research Committee also worked with the Waukesha Motor
Company to develop a standardized CFR fuel research engine and specific operating
conditions to measure the octane number of a fuel. The first CFR engine was designed
and built in 1929, and it is still the standard by which liquid fuel octane measurements
are made today. It is a single-cylinder (3.25 in. bore and 4.50 in. stroke) four-stroke valve
in-head engine. The cylinder head and sleeve are one piece, so the compression ratio can
be varied from 3:1 to 15:1 by lowering the entire cylinder with respect to the piston. It
usually has a shrouded intake valve to induce swirl and turbulence, and to promote mixing.
The CFR engine has no valve overlap, so the intake valve does not open substantially until
the piston is about 1/3 to 1/2 of the way down the cylinder. The engine is coupled to a
synchronous electric generator that is used to control the engine speed to a submultiple of
the line frequency.
To measure knock, an ASTM pressure pickup that responds to the rate of pressure rise
is used. The pickup consists of a core rod of magnetostrictive alloy. As the pressure rises
in the combustion chamber, the diaphragm transmits this force to the core rod that in turn
generates a magnetic field. The copper wire coil around the core converts the magnetic
field to a voltage that is proportional to the rate of change of the combustion pressure.
An evaporative cooling system is used to maintain constant cylinder jacket temperature.
The coolant vaporizes in the cylinder jacket, with the vapor flowing to a condenser, and
recirculates back to the cylinder.
Two sets of CFR engine operating conditions for engines are employed: the research
(ASTM D908) and the motor (ASTM D357) methods, as detailed in Table 7.3. The table
also includes specifications (ASTM D614) for testing aviation engines. Originally, the
octane number measurement was performed using the research method; however, this
method did not correlate well with the knocking found from actual road conditions at wide
open throttle. The motor method, with increased engine speed and spark advance, was
Table 7.3 Octane Number Measurement Conditions
ASTM method
Air inlet temperature (K)
Jacket temperature (K)
Speed (rpm)
Spark advance (degrees btdc)
Research
Motor
Aviation
D908
288
373
600
13
D357
310
373
900
19--26
D614
325
463
1200
35
Abnormal Combustion (Knock) in Spark Ignition Engines
213
developed to give an improved correlation between road and laboratory knocking results.
The octane number label on gasoline pumps is the average of the research (𝑅) and the
motor (𝑀) method octane numbers, (𝑅 + 𝑀)∕2, and is also called the antiknock index
(AKI).
The procedure to measure the octane number of a test fuel is as follows:
1. Run the CFR engine on the test fuel at either the motor or the research operating
conditions.
2. Slowly increase the compression ratio until the standard amount of knock occurs.
3. At that compression ratio, run the engine on blends of the reference fuels isooctane and
𝑛-heptane.
4. The octane number is the percentage of isooctane in the blend that produces the standardized knock at that compression ratio.
One measure of an engine’s octane requirement is its knock-limited indicated mean effective
pressure (klimep). The greater the knock-limited imep, the smaller the octane requirement.
Knock-limited imep is measured by increasing the inlet pressure 𝑃i (which will increase the
cylinder charge density and temperature) until knock occurs; the imep at that condition is
the knock-limited imep. Experimental results (Hesselberg and Lovell, 1951) indicate that
klimep decreases with increasing coolant temperature. Similar results are obtained with
increasing inlet air temperature. Both results are to be expected, since chemical reaction
rates are accelerated strongly by increase in temperature.
There are two problems with the octane number scale:
1. At low coolant temperatures, di-isobutylene performs better than isooctane (implying
the octane number is greater than 100).
2. The relative ranking of isooctane and di-isobutylene depends on coolant temperatures.
If the octane scale were decoupled from engine design, making the assigned number a
fuel property, the fuel with the greater octane number would always yield the largest
klimep.
3. Alcohol fuels have an octane number greater than 100.
The former problem is dealt with by extrapolation. A performance number defined as
the ratio of the knock-limited imep for the fuel in question to the knock-limited imep of
isooctane is used for this purpose. The latter problem is dealt with by using two standard
operating conditions (research and motor) and reporting an average number. These shortcomings should be kept in mind; they are easy to forget because of the great utility of the
octane number scale.
Typical results obtained for the effect of fuel--air ratio on the knock-limited imep are
shown in Figure 7.14. Notice that near-stoichiometric mixtures have the lowest klimep
(therefore the highest octane requirement). Also, notice that maximum klimep is attained
with very rich (𝜙 ∼ 1.6) mixtures. Therefore, to obtain maximum power from an engine,
one should run very rich, near 𝜙 ∼ 1.6, with a compression ratio and inlet pressure such
that imep is equal to knock-limited imep.
Since knock occurs if there is enough time for sufficient autoignition precursors to
form, at high engine speeds one might not expect knock to be a problem since there is less
time available for the precursors to form. On the other hand, as engine speed increases,
there is less heat transfer from the gases to the cylinder walls so that gas temperatures
will be higher. This accelerates the precursor formation rate so that less time is required to
form a concentration high enough for autoignition to occur. As a result of these and other
214
Combustion Processes in Engines
15
1
Knock-limited imep (bar)
14
13
2
12
3
11
10
9
8
Figure 7.14 Effect of fuel--air ratio on
knock-limited imep for three aircraft
fuels (Cook et al., 1944).
7
0.8
1.0 1.2 1.4 1.6
Equivalence ratio
1.8
competing effects, some engines show a klimep increasing with speed, and others show a
decrease.
Knock in gaseous--fueled spark ignition engines is characterized with a methane number (MN), (Leiker et al., 1972), which is the percentage by volume of methane blended with
hydrogen that exactly matches the knock intensity of the test gas mixture under specified
operating conditions in a CFR engine. For example, a blend of 20% hydrogen and 80%
methane has a methane number (MN) of 80. For the range beyond 100 MN, methane-carbon dioxide mixtures are used as reference mixtures. In this case, the MN is 100 plus
the percent CO2 by volume in the reference methane--carbon dioxide mixture. In order to
replicate the ASTM D357 motor method, testing is conducted at the air--fuel ratio AF that
produces maximum audible knock.
7.4 COMBUSTION IN COMPRESSION IGNITION ENGINES
Combustion Diagnostics
To better understand compression ignition combustion processes, we first discuss flow
visualization and diagnostic techniques used in compression ignition engines. The application of optical diagnostics to diesel combustion is constrained by the need to maintain
realistic combustion chamber geometry while maintaining satisfactory optical access.
High-speed cinematography is a qualitative measurement technique, since the image
is integrated along the line of sight of an optically thick medium, and it is also not species
specific, so additional diagnostic techniques have been developed. Various laser-based
combustion diagnostic techniques developed by combustion researchers have been applied
to diesel engines in order to obtain more detailed and species specific information about
the combustion processes taking place in a diesel engine. The techniques include laser light
scattering, and laser-induced incandescence and fluorescence.
With a light scattering technique, laser light is elastically scattered by fuel droplets
and/or soot particles. The scattering distribution and intensity depends on the particle
size. Mie scattering, named after Gustav Mie (1869--1957), is defined as elastic scattering
Combustion in Compression Ignition Engines
215
from particles whose diameter is of the same order of magnitude or smaller as the light
wavelength. Liquid droplet spray patterns have been determined via measurements of the
Mie scattering. Elastic scattering of light from molecules or small particles with diameters
much smaller than the wavelength of the laser light is termed Rayleigh scattering, named
after Lord Rayleigh (1842--1919). Vaporized fuel--air mixture patterns and temperature
fields are determined through Rayleigh scattering measurements.
Laser light is also used to induce incandescence and fluorescence of given species.
Both relative and absolute soot concentrations have been determined using laser-induced
incandescence (LII). Planar laser-induced fluorescence (PLIF) has been used to determine
polyaromatic hydrocarbon (PAH) concentrations, which are precursors to soot, OH distributions, and NO distributions. The OH radical distribution provides information about
the location and intensity of diffusion and premixed flames. The NO radicals indicate the
location of NO𝑥 production in the cylinder.
An optical engine with an extended piston and a piston-crown window optical access
was originally introduced by Bowditch (1961). A more recent example is a single-cylinder
diesel engine modified for optical access by researchers at Sandia National Laboratory (Dec
and Espey, 1995). This direct injection research engine is based on a typical commercial,
heavy-duty diesel engine, with a stroke of 140 mm and a bore of 152 mm. It has an extended
piston with a piston crown and a window at the top of the cylinder to provide for laser
access along the axis of the fuel spray. The combustion bowl has a flat ‘‘pancake’’ bottom
allowing the laser sheet to be viewed from above and below throughout the bowl and the
squish region. The in-cylinder flow is quiescent with no swirl from the incoming airflow,
and the test diesel fuel used in the engine had a cetane number of 42.5.
Figure 7.15 shows a high-speed cinematography sequence of the luminous combustion process in the above test engine. The high-speed photo sequence shows the fuel jet
penetration and spread of the luminous combustion zones. The start of injection (SOI) is
at 8.8◦ btdc, and each image is about 2.4◦ apart. The first luminosity is seen about 5◦ after
the start of injection. The rapid appearance of widespread combustion indicates that the
ignition occurs at multiple points throughout the jet. The luminosity is yellow, indicating
the presence of hot soot particles and suggesting fuel-rich combustion. Between 2.9◦ and
1.8◦ btdc, the burning fuel jets contact the edge of the combustion bowl, and then spread
along the circumference into the space between the jets and downward into the bottom of
the bowl. The combustion heat release ends at about 17◦ atdc. For this case, about half of
the fuel is injected after the burning fuel jet reaches the edge of the bowl. Since this is a
quiescent engine, a significant portion of the in-cylinder air is not utilized for sometime
after SOI, and as the injection proceeds, the fuel spray must travel ever farther through
burned gases where combustion has already occurred.
Diesel Combustion Process
In a diesel engine, a low volatility fuel must be converted from a liquid state into a
finely atomized state, vaporized, mixed with air, and its temperature raised to a point to
support autoignition. The diesel combustion process is heterogeneous and controlled by the
rate of fuel--air mixing. It has been classified into three phases: ignition delay, premixed
combustion, and mixing-controlled combustion.
The time interval between the start of injection and the start of combustion is termed
the ignition delay. Once regions of vapor--air mixture formed around the fluid jet as
it is first injected into the cylinder are at or above the autoignition temperature, they
will spontaneously ignite. The combustion of this initial vapor--air mixture is termed the
216
Combustion Processes in Engines
Figure 7.15 High-speed photographic sequence of the luminosity of a diesel flame (Espey and
Dec, 1993).
premixed combustion phase. In the premixed combustion phase, ignition and combustion
occurs around the fuel jet in regions that are fully mixed.
Next, in the mixing-controlled combustion phase, the fuel entering the cylinder does
not mix fully with the air before combusting but burns in what is termed a diffusion flame.
In a diffusion flame, the fuel in the main body of the fuel jet mixes with the surrounding
air and ignites over a narrow range of equivalence ratios. Combustion in this phase occurs
at a rate limited by the rate at which the fuel can be mixed with the entrained air.
Combustion in Compression Ignition Engines
217
Figure 7.16 Simple model of
diesel combustion.
The quantity of fuel burned in each of the premixed and mixing-controlled phases is
not only influenced by the engine and injector design, but also by the fuel type and the load.
At idle, most of the fuel injected in small bore diesel engines is burned in the premixed
phase. As the load increases, the injection duration increases, and the relative size of the
mixing-controlled phase increases relative to the premixed phase.
Early analyses of diesel combustion assumed that a burning diesel jet was composed
of a dense fuel-rich core surrounded by a uniformly leaner fuel--air mixture, as shown in
Figure 7.16. With reference to the models used for steady spray combustion in furnaces
and gas turbines, the diesel fuel autoignition and premixed combustion phases were also
assumed to occur in a diffusion flame in the near stoichiometric (𝜙 ∼ 1) regions between
the rich (𝜙 > 1) and the lean (𝜙 < 1) limits, at the outer edge of the jet. Soot was assumed
to form in a narrow region on the fuel-rich side of the diffusion flame.
Recent laser sheet diagnostic experiments in diesel engines have indicated that the
combustion process in diesel engines is different than that in furnaces and gas turbines. Dec
(1997) has proposed an alternative conceptual model based on laser sheet experimental
results. The Dec model features two stages of fuel oxidation for both of the premixed and
mixing-controlled combustion phases. The first stage is partial oxidation of the fuel in a rich
premixed reaction, and the second stage is combustion of the fuel-rich, partially oxidized
products of the first stage in a near stoichiometric diffusion flame.
This conceptual model is shown schematically in Figure 7.17, a temporal sequence
showing the progressive changes during the injection process. Significant events in the
evolution of the jet state are drawn at successive degrees after the start of injection (ASI).
Six parameters are shown in Figure 7.17: the liquid fuel, the vapor--air mixture, the PAHs,
the diffusion flame, the chemiluminescence emission region, and the soot concentration.
At 1.0◦ in Figure 7.17, near the beginning of the ignition delay phase, as the liquid fuel
is injected into the cylinder, it entrains hot cylinder air along the sides of the jet, leading
to fuel evaporation. Note that throughout the injection process, the liquid length portion
of the jet remains relatively constant. There is limited penetration of the fuel droplets into
218
Combustion Processes in Engines
1.0
ASI
5.0
ASI
2.0
ASI
6.0
ASI
3.0
ASI
6.5
ASI
4.0
ASI
8.0
ASI
4.5
ASI
10.0
ASI
0
10
20
Scale (mm)
Liquid fuel
Vapor_fuel/air mixture
(equivalence ratio 2:4)
Low
High
PAHs
Soot concentration
Diffusion flame
Chemiluminescence
emission region
Figure 7.17 Detailed model of diesel combustion (Dec, 1997).
the combustion chamber. The penetration depth of the liquid jet has been found to be
dependent on the volatility of the fuel, injector hole size, fuel and cylinder air temperature,
and relatively insensitive to the injection pressure (Siebers, 1998).
At about 4.0◦ ASI, a vapor head vortex is beginning to form in the leading portion of the
jet downstream of the liquid jet. The bulk of the vaporized fuel is in the head of the jet. The
fuel vapor--air mixture region in the head vortex is relatively uniform, has a well-defined
boundary separating it from the surrounding air, and has an equivalence ratio between 2 and
4 throughout its cross section. At about 5.0◦ ASI, premixed combustion begins in the head
vortex. As a consequence of the high equivalence ratio, the initial premixed combustion
is fuel-rich with a temperature of about 1600 K and produces PAHs and soot. The soot
concentration is fairly uniform throughout the jet cross section.
At about 6.5◦ ASI, a turbulent diffusion flame forms at the edge of the jet around the
products of the initial premixed stage. This turbulent diffusion flame begins the transition
to the mixing-controlled phase and is near stoichiometric. The diffusion flame causes the
formation of larger soot particles at the jet periphery. The soot concentration continues to
increase throughout the head vortex region at the head of the jet. Since the head vortex of
the jet is composed of recirculating gases, the soot particles also recirculate and grow in
size.
Combustion in Compression Ignition Engines
219
At about 8◦ ASI, the jet reaches a quasi-steady condition in which the general features
of the jet do not change significantly as it expands across the combustion chamber. The
combustion is in the mixing-controlled phase. The fuel first passes through a very fuel
rich (𝜙 > 4) premixed reaction stage and then burns out in the turbulent diffusion flame
at the edge of the jet. Most of the soot is burned with the fuel in the diffusion flame.
The fraction of soot that is not oxidized becomes an exhaust emission. NO𝑥 is formed in
the high-temperature regions in the diffusion flame where both oxygen and nitrogen are
available, and in the post combustion hot gas regions.
Diesel Cetane Number
Diesel fuels are compared using an ignition delay metric and classified by cetane number
(CN). The cetane number characterizes the ability of the fuel to autoignite, the opposite of
octane number. The higher the cetane number, the shorter the ignition delay, as the ignition
delay decreases approximately linearly with cetane number. If the cetane number is too
low, the fuel will not ignite until late in the injection process. In this situation, the fuel is
well mixed so that once combustion is initiated, the burning rate is very high, causing diesel
knock to occur. At higher cetane numbers, combustion is initiated while the fuel is being
injected, so the burning rate is controlled by the rate of fuel--air mixing. Cetane numbers
for vehicular diesel fuels range from about 40 to 55. Additives such as nitrate esters can be
used to increase the cetane number.
The cetane number scale varies from 0 to 100 and is measured for a given test fuel
using a standard CFR engine with a prechamber and a variable compression ratio, and
operated according to a standard set of operating conditions (ASTM D613) shown in Table
7.4. The compression ratio is adjusted until the ignition delay is 13◦ with the test fuel. At
that compression ratio, reference fuels are blended to again produce an ignition delay of
13◦ . The cetane number is then computed from the relation below:
CN = % hexadecane + 0.15 × (% heptamethyl nonane)
The name cetane is derived from the fact that hexadecane is referred to as 𝑛-cetane
(C16 H34 ). The cetane number of 𝑛-cetane is assigned a value of 100. Originally, the cetane
scale assigned a value of zero to a-methylnaphthalene as a reference fuel. Later, the lowcetane reference fuel was changed to heptamethylnonane (HMN), as it is less expensive
and has better storage characteristics, and assigned a cetane number of 15 so that results
obtained in the past were still valid. The cetane number and the octane number are inversely
correlated, as shown in Figure 7.18. Gasoline is a poor diesel fuel and vice versa.
A low cetane number will mix more completely with the cylinder air before burning
so that the local equivalence ratio of the initial premixed burn will be less (𝜙 ∼ 3) than the
local equivalence ratio (𝜙 ∼ 4) for a greater cetane number.
Table 7.4 Cetane Number Measurement Conditions (ASTM D613)
Inlet temperature (◦ C)
Coolant temperature (◦ C)
Speed (rpm)
Injection timing (btdc)
Injection pressure (MPa)
66
100
900
13◦
10.3
220
Combustion Processes in Engines
120
115
105
100
95
90
Motor method octane number
110
80
70
60
40
0
–20
0
20
40
60
80
100
120
Cetane number
Figure 7.18 Cetane and octane number correlation for hydrocarbon fuels. (Adapted from Taylor,
1985.)
Diesel Ignition Delay
Diesel ignition delay is a physical process involving both fluid mechanics and combustion.
During this process, the injected fuel is atomizing into droplets, vaporizing, mixing with
the entrained air, and initiating preflame reactions. Accordingly, the diesel ignition delay
period is defined as the time or crank angle between the start of injection (SOI) and the start
of combustion (SOC). The start of injection begins when the injector needle lifts off its seat
and fuel begins to flow into the combustion chamber. The start of combustion is determined
indirectly using combustion analysis. One SOC metric is the crank angle location where
10% of the total energy release is reached. A similar SOC metric is the crank angle location
where the rate of change of the slope of the cylinder pressure profile is a maximum.
The ignition delay period 𝜏id depends not only on the chemical characteristics of the
fuel, but also on the fluid mechanics of atomization, vaporization, and mixing. Aromatic
hydrocarbons and alcohols have chemical bonds that are difficult to break and result in a
long ignition delay. If these fuels are injected rapidly enough to mix completely with air
before autoignition occurs, they will all burn rapidly when ignition occurs in the premixed
phase, producing a large rate of change of pressure and a high peak pressure.
On the other hand, the chemical bonds of some fuels, such as alkanes (straight chain
paraffins), are easily broken. Ignition delay is then short, and with a long injection, most
of the fuel to be burned is injected after autoignition occurs. Relatively little fuel burns in
Combustion in Compression Ignition Engines
221
the premixed phase and most of the fuel burns at a rate limited by the rate of mixing with
the cylinder air.
As the engine speed increases, the ignition delay period will need to decrease to
maintain a relatively constant combustion duration in crank angle degrees. In direct injection
engines, the fuel--air mixing rate is increased using increased turbulence, swirl generated
by the intake port geometry, and deeper piston bowls.
There have been a number of empirical correlations that have been developed for
ignition delay that include the cetane number CN, the cylinder pressure 𝑃 , temperature 𝑇 ,
and mean piston speed 𝑈̄ p . A correlation, Equation 7.16, by Hardenberg and Hase (1979)
for direct injection engines is widely referenced, where the ignition delay is in crank angle
degrees, piston speed in m/s, pressure in bars, and temperature in degrees Kelvin:
[ (
)(
)0.63 ]
21.2
1
1
̄
−
(7.16)
𝜏id = (0.36 + 0.22𝑈p ) exp 𝐸a
𝑅u 𝑇
17, 190
𝑃 − 12.4
The term 𝐸a is the apparent activation energy in J/mole,
618, 840
(7.17)
CN + 25
and the temperature and pressure are computed at top dead center assuming a polytropic
process with exponent 𝑛 and compression ratio 𝑟:
𝐸a =
𝑇 = 𝑇tdc = 𝑇i 𝑟𝑛−1
𝑃 = 𝑃tdc = 𝑃i 𝑟𝑛
EXAMPLE 7.2
(7.18)
Diesel Engine Ignition Delay
A diesel engine with a stroke of 165 mm and compression ratio of 18 operates at 1500 rpm
using a diesel fuel with a cetane number of 40. Given that the inlet manifold temperature and
pressure 𝑇i = 283 K, 𝑃i = 1.0 bar, and 𝑛 = 1.35, what is the estimated ignition delay (ms)?
SOLUTION
𝑈̄ p = 2𝑁𝑠 = (2)(1500∕60)(0.165) = 8.25 m∕s
𝑇 = 𝑇i 𝑟𝑛−1 = (283)181.35−1 = 780 K
𝑃 = 𝑃i 𝑟𝑛 = (1)181.35 = 49.5 bar
618, 840
𝐸a =
= 9521 J
𝐶𝑁 + 25
so the ignition delay is
[ (
)(
)0.63 ]
21.2
1
1
−
𝜏id = (0.36 + 0.22𝑈̄ p ) exp 𝐸𝐴
𝑅u 𝑇
17, 190
𝑃 − 12.4
[
(
)(
)0.63 ]
21.2
1
1
= (0.36 + 0.22(8.25)) exp (9521)
−
(8.314)(780) 17, 190
49.5 − 12.4
= 4.1 deg
and in terms of time,
𝜏id = 4.1 deg (60 s/min) (rev/360 deg) (1000 ms/s)/(1500 rev/min) = 0.45 ms
222
Combustion Processes in Engines
Note that Equation 7.16 predicts that the ignition delay in crank angle degrees increases
linearly with engine speed at a constant load, and that the ignition delay in terms of time is
relatively constant.
For additional considerations regarding diesel ignition delay, the reader is referred to
the paper by Assanis et al. (2003).
Diesel Energy Release Analysis
Diesel engine combustion analysis is performed using a differential energy equation analysis to determine either the effective energy release or the effective fuel injection rate for
a given cylinder pressure profile. Typical calculations for a direct injection (DI) engine
use Equation 7.20, which assumes homogeneous conditions throughout the combustion
chamber during the injection and combustion process, and ideal gas behavior. The energy
release in indirect injection engines (IDI) is modeled with an energy equation applied to
both the main chamber and the prechamber so that pressure data are required for both
chambers. Diesel energy release profiles will typically have two maxima, resulting from
the premixed and the mixing-controlled combustion phases.
The double peak shape of the energy release profile in Figures 7.19 is characteristic of
diesel combustion. The first peak occurs during the premixed combustion phase and results
from the rapid combustion of the portion of the injected fuel that has vaporized and mixed
with the air during this period. The energy release curve in the premixed combustion phase
is relatively independent of the load, since the initial mixing is independent of the injection
duration. The second peak occurs during the mixing-controlled combustion. The energy
release during this phase depends on the injection duration. As the injection duration is
increased to meet an increased engine load, the amount of fuel injected increases, thus
increasing the magnitude and duration of the mixing-controlled energy release.
A dual Wiebe function (see Figure 7.20), which has two peaks, has been used to
fit diesel combustion energy release data (Miyamoto et al., 1985). The dual equation,
Apparent heat release rate (J/deg)
240
Short injection
200
Long injection
160
120
80
40
0
–40
–20
–10
0
10
20
Crank angle (degrees atdc)
30
40
Figure 7.19 Example energy release profile for short- and long-fuel injection. Adapted from Dec
(1997).
Combustion in Compression Ignition Engines
223
Rate of combustion
Premixed combustion
Mixing-controlled
combustion
Figure 7.20 Dual Wiebe function for
diesel energy release. Adapted from
Miyamoto (1985).
Crank angle
Equation 7.19 with seven parameters is
𝑄p
𝑑𝑄
= 𝑎 𝑚p
𝑑𝜃
𝜃p
(
𝑄
+ 𝑎 𝑚d d
𝜃d
𝜃
𝜃p
(
)𝑚p −1
𝜃
𝜃d
[
exp −𝑎
)𝑚d −1
(
[
exp −𝑎
𝜃
𝜃p
(
)𝑚p ]
𝜃
𝜃d
)𝑚d ]
(7.19)
The subscripts p and d refer to the premixed and mixing-controlled combustion portions, respectively. The parameter 𝑎 is a nondimensional constant, 𝜃p and 𝜃d are the burning
duration for each phase, 𝑄p and 𝑄d are the integrated energy release for each phase, and
𝑚p and 𝑚d are nondimensional shape factors for each phase. The adjustable parameters are
selected using a least squares fit. Miyamoto et al. (1985), for the specific direct (DI) and
indirect injection (IDI) diesel engines tested in their experiments, reported that the 𝑚p , 𝑚d ,
and 𝜃p parameters were essentially independent of engine speed, load, and injection timing.
The fitted values of these parameters is 𝑎 = 6.9, 𝑚p = 4, 𝑚d = 0.5 (DI) or 0.9 (IDI), and
𝜃p = +7◦ .
The effective diesel fuel injection rate can also be obtained using the energy equation.
The effective fuel injection rate 𝑚̇ f is based on the assumptions that the chamber mixture
is homogeneous and in thermodynamic equilibrium. The different liquid and vapor fuel
fractions are not included at this level of modeling. The open system first law for the
combustion chamber, Equation 7.20 with the injected fuel now explicitly included is
𝑑
−𝑄̇ l − 𝑃 𝑉̇ = (𝑚𝑢) − 𝑚̇ f ℎf
𝑑𝑡
(7.20)
and the mass conservation equation, Equation 7.21 is
𝑑𝑚
= 𝑚̇ f
𝑑𝑡
(7.21)
In Equations 7.20 and 7.21, 𝑚 is the fuel--air mass in the cylinder, 𝑚̇ f is the fuel injection
rate, ℎf is the enthalpy of the injected fuel, and 𝑄̇ l is the heat transfer loss rate. With the
above assumptions, the ideal gas equation in differential form is
𝑃 𝑉̇ + 𝑉 𝑃̇ = 𝑅 𝑇 𝑚̇ + 𝑅 𝑚 𝑇̇
(7.22)
Combustion Processes in Engines
If dissociation is neglected, the internal energy is a function of temperature, pressure, and
equivalence ratio only, so
(7.23)
𝑢 = 𝑢(𝑇 , 𝑃 , 𝜙)
Differentiation of Equation 7.23 with respect to time gives
𝜕𝑢 ̇
𝜕𝑢 ̇
𝜕𝑢 ̇
𝑇+
𝑃+
𝜙
𝜕𝑇
𝜕𝑃
𝜕𝜙
𝑢̇ =
(7.24)
If the mass of air in the cylinder is constant, with no residual fuel in the chamber at
the beginning of injection, the overall equivalence ratio increases solely due to the fuel
injection, and in differential form is
𝜙̇ = 𝜙
𝑚̇ f
𝑚f
(7.25)
Finally, combining Equations 7.20 through 7.25 leads to
)
(
𝑐
𝑐
−𝑄̇ l − 1 + 𝑅v 𝑃 𝑉̇ − 𝑅v 𝑉 𝑃̇
𝑚̇ f =
𝜕u
𝜙
𝑢 − ℎf − 𝑐v 𝑇 + 𝑚𝑚 𝜕𝜙
(7.26)
f
Equations 7.22, 7.25, and 7.26 are a set of ordinary, differential equations that when
̇ 𝑚f and
numerically integrated using measured values for 𝑃 , 𝑃̇ , 𝑉 , and 𝑉̇ yield 𝑇 , 𝑇̇ , 𝜙, 𝜙,
𝑚̇ f as functions of time. At each time step, an equilibrium combustion product numerical
routine gives the required partial derivatives of the internal energy. The heat transfer loss
𝑄̇ l is computed at each time step from an appropriate model.
Results obtained by an energy release computation that includes the relatively small
effects of dissociation are given in Figure 7.21. The cylinder pressure and effective fuel
injection rate (mg/deg) are plotted as a function of crank angle. The effective fuel injection
rate curve is double peaked, similar to the effective energy release rate. The area under the
curve is approximately equal to the actual mass of fuel injected.
90
60
50
40
30
tdc
20
–20 –10 0 10 20 30 40
Crank angle (deg)
16
Apparent rate of burning (mg/deg)
70
8
7
6
5
4
3
2
1
0
–1
–2
–3
Pressure derivative (bar/deg)
80
Cylinder pressure (bar)
224
14
12
10
8
6
4
2
0
–20 –10
tdc
0
10
20
30
Crank angle (deg)
40
Figure 7.21 The effective fuel injection rate versus crank angle. Adapted from Krieger and
Borman (1966).
Low-Temperature Combustion
225
Multidimensional Numerical Models of Diesel Combustion
Since diesel combustion is heterogeneous, numerical models need to be multidimensional
to account for the spatial variations in temperature and species concentrations. There are
a variety of turbulent combustion models, including probability density function (pdf)
models, eddy break-up models, flamelet models, and coherent flame models that are used
for both spark and diesel combustion modeling.
The research code KIVA (Amsden et al., 1985), developed at Los Alamos National
Laboratory, is a public domain three-dimensional CFD program that has been used by a
number of research groups to model compression ignition combustion. Reitz and coworkers
at the University of Wisconsin (for example, see Kong et al., 1995) have added a number
of improvements to the original KIVA model that incorporate more realistic analysis of
the fuel spray breakup, vaporization, spray-wall impingement, wall heat transfer, ignition,
combustion, and pollutant formation.
7.5 LOW-TEMPERATURE COMBUSTION
Introduction
Recently, there has been a great deal of interest and research activity worldwide in lowtemperature combustion, in which combustion is initiated by the autoignition of a lean
and nearly homogeneous fuel--air mixture. Over the last 20 years, a variety of techniques,
categorized as ‘‘low-temperature combustion’’, have been developed to combine the best
aspects of spark and diesel combustion, that is, combining the homogeneous fuel--air
mixture of a spark ignition engine with the higher compression of a diesel engine to
achieve lower emissions with much higher engine efficiencies.
The combustion process in a spark ignition engine produces less emissions than that
of a diesel engine, but its compression ratio is knock-limited, and it has increased pumping
losses at part load. The diesel engine is more fuel efficient, since it operates at a higher
compression ratio and is unthrottled, but since it has higher NO𝑥 and particulate (soot)
emissions, it can require exhaust after treatment to meet emission standards.
In an LTC engine, the combustion begins whenever the autoignition temperature of
a homogeneous or near homogeneous fuel--air mixture is reached. The low combustion
temperatures greatly reduce the formation of nitric oxides and the heat transfer losses,
and the lean air--fuel ratios reduce soot formation. Over the last decade, the hardware and
software technology for control and operation of low-temperature combustion has advanced
to a level that allows greater use of this combustion process. These techniques have taken
advantage of the advances in injection technology, such as common rail injectors, allowing
precise control of injection timing, multiple injection events, and use of multiple fuels.
An advantage of LTC is fuel flexibility, as both liquid fuels (gasoline, diesel, and
biodiesel) and gaseous fuels (natural gas and hydrogen) can be used. The reactivity of
the fuel is used as an ignition metric. Low reactivity fuels such as gasoline that have a
high octane number are resistant to autoignition, and high reactivity fuels such as diesel
fuel that have a high cetane number are more susceptible to autoignition. In addition to
ignition delay, considerations such as fuel volatility, fuel composition, and initial cylinder
temperature and pressure are important.
A volatile low reactivity fuel will vaporize and mix with the intake air more rapidly but
will require higher cylinder temperatures for autoignition. A less volatile high reactivity fuel
will require earlier injection for vaporization and mixing at a lower cylinder temperature.
As the fuel molecular size increases, for example, from pentane to hexane and heptane, the
226
Combustion Processes in Engines
ignitability and rate of energy release increases, since the longer carbon chains break up
more easily into radicals.
For the engine designer, with a low-temperature combustion approach, the energy
release profile can be tailored to different modes of operation and fuel mixtures to meet
emission requirements over the load and speed conditions of an engine. During lowtemperature combustion, there is no distinct flame or wave front that propagates across the
chamber but simultaneously ignites at a number of sites throughout the cylinder. With this
propagation of ignition sites, the energy release is volumetric. The combustion duration is
shorter, and the peak pressures and energy release rates are greater relative to conventional
combustion. Soot production is greatly reduced, since the mixture is nearly homogeneous,
with no locally rich combustion zones.
However, since the combusting mixture is nearly homogeneous, the start of ignition
and the reaction rate is controlled by chemical kinetics, not turbulent diffusion as it is
with non premixed combustion. Therefore, the cylinder temperature, pressure, equivalence
ratio, and fuel composition govern the start of ignition. The low-temperature combustion
duration can be very short, typically 5--10◦ , compared with 20--40◦ for a conventional
spark or diesel engine, with an accompanying high rate of energy release.
The issues with low-temperature ignition that arise in engines are the high-energy
release rates, adequate mixing of the fuel, air, and residual gases at high engine speeds,
transient operation, cold starting, control of the start of ignition and the duration of ignition
over a range of engine speeds and loads, and increased HC and CO emissions due to partial
fuel oxidation. In response, researchers have developed a number of low-temperature
combustion strategies to address these problems.
Three representative techniques that have been developed are homogeneous charge
compression ignition (HCCI), premixed charge compression ignition (PCCI), and
reactivity-controlled compression ignition (RCCI). In the latter two techniques, some
degree of charge stratification is required to prevent excessive energy release rates and
pressure rise. The charge stratification is accomplished using exhaust gas recirculation and
multiple fuel injection.
Homogeneous Charge Compression Ignition (HCCI)
One of the first low-temperature combustion techniques developed was homogeneous
charge compression ignition (HCCI). HCCI research began in the late 1970s, for example,
see Onishi et al. (1979) for two-stroke engines and Najt and Foster (1983) for four-stroke
engines. The two-stroke HCCI engines used very high EGR of up to 80% in which the
goal was to improve combustion stability and reduce fuel consumption at part load. An
HCCI engine operates without throttling, reducing the pumping losses, and the load is met
by the control of the fuel--air ratio, so the mixture is very lean at low loads, and as the
load increases, the mixture becomes more stoichiometric. The lean (𝜙 < 0.3) mixture in an
HCCI combustion process results in lower peak combustion temperatures, about 1300 K,
in comparison to spark ignition and diesel engines, depending on the fuel.
The lower peak temperatures significantly reduce NO𝑥 . However, HCCI engines have
an upper limit on the fuel--air ratio and peak torque. With an increase in load, there is
an increase in the peak combustion temperature, and corresponding increase in the NO𝑥
levels and susceptibility to knock. Also, as the load is increased, more fuel is added,
increasing the fuel--air ratio, and thus advancing the start of combustion, increasing the
net compression work. Experiments have shown there is a knock limit at higher loads, and
partial combustion at high engine speeds. Since there are also lean flammability limits at
Low-Temperature Combustion
227
low load, the operating range of HCCI can be relatively narrow and limited to midload
conditions.
The knock in an HCCI engine is a volumetric knock, in which autoignition is initiated
simultaneously near TDC at multiple locations in the cylinder. Measurements by Vavra et
al. (2012) indicate that the dominant frequency in HCCI knock is the first tangential (1T)
mode with a uniform decrease in amplitude to other oscillation modes. This is in contrast
to SI end gas knock which was found to begin after TDC near a wall with a greater energy
density, more stochastic with less uniform distribution in frequencies, a larger difference
in knock magnitude from the 1T to the 2T mode, and more cycle--cycle behavior.
In an HCCI engine, since the start and duration of combustion can not be directly
controlled, it is controlled indirectly. Control techniques include varying inducted gas temperature, fuel--air ratio, high EGR rates, variable compression ratio, and variable valve
timing. Using a variable compression ratio will control the temperature rise during compression. The residual fraction can be controlled using variable valve timing, for example,
using early exhaust valve closing will increase the residual fraction.
The implementation of HCCI is different for gasoline and diesel fuels due to their
different ignition characteristics. Dec and Yang (2010) used gasoline as a fuel for a medium
duty diesel engine with a compression ratio of 14:1 operating in a HCCI mode, and reported
that by boosting the intake pressure from 100 kPa to 325 kPa absolute while also increasing
the EGR from 0 to 60%, they were able to increase the IMEP from 4 to 16 bar at the
knock/stability limit. The gasoline entered the cylinder fully premixed in the intake plenum,
eliminating fuel--air mixing issues.
Conversely, with diesel fuel, there is a need to cool the intake air to prevent knock
and to start the fuel injection earlier in the compression stroke to give enough time for a
homogeneous mixture to form. In this case, cooled EGR or reduced compression ratios are
used to increase the ignition delay and decrease the rate of energy release. One method used
to achieve adequate mixing in diesel-fueled LTC engines is early injection. However, the
low volatility of diesel fuel can result in wall wetting, so low-penetration fuel injectors are
used to increase entrainment and reduce wall impingement. The low penetration is achieved
with higher pressure fuel injection together with an increased number of holes that have a
smaller diameter. Multiple injection is also used to tailor the injection into several pulses
with different durations. Also, with early injection, since the piston is lower in the cylinder,
the fuel spray needs to have a larger downward direction.
Partially Premixed Compression Ignition (PPCI)
The partially premixed compression ignition (PPCI) technique is used to increase the
combustion duration and reduce the rate of pressure rise through partial mixing of the fuel
to create fuel stratification in the cylinder. With this technique, multiple injections of a
single fuel are employed. The mixture is thus stratified, with a distribution of equivalence
ratio. With a two injection strategy, as illustrated in Figure 7.22, a pilot injection is placed
early in the compression stroke, typically about 150◦ btdc to create a homogeneous mixture,
with the main injection near top dead center. Three injection event schemes have also been
used. Combustion is initiated by the main injection near top dead center. High levels of
EGR are also used to increase the ignition delay and prevent combustion during the first
injection.
The relative amounts of fuel used in each injection depend on the engine load, fuel
reactivity, and level of EGR. Both gasoline and diesel fuels have been used successfully in
engines operating in a PPCI mode (Kalghatgi, 2007). A variation of PPCI (Dec et al., 2011)
228
Combustion Processes in Engines
Injector flow rate (mg/ms)
25
Figure 7.22 Example PPCI dual
injection strategy.
Main injection
20
15
10
Pilot injection
5
0
–150
–100
–50
0
50
Crank angle (degrees atdc)
100
is to premix the portion of the fuel with the intake air followed by directly injecting the
remaining portion of the fuel into the cylinder during compression to initiate combustion.
Ra et al. (2012), using a PCCI strategy with a triple injection of gasoline into a light duty
1.9 L single-cylinder diesel engine with a 16.5:1 compression ratio operating at about 16
bar and 2500 rpm, measured an indicated specific fuel consumption as low as 172 g/kWh.
The EGR level was 48%.
Reactivity Controlled Compression Ignition (RCCI)
The reactivity-controlled compression ignition (RCCI) technique uses a dual fuel compression ignition strategy. As the schematic in Figure 7.23 illustrates, RCCI begins with
port injection of a low reactivity (low cetane number) fuel such as gasoline followed by
multiple direct injection of relatively small amounts (≃ 15%) of a high reactivity (high
cetane number) fuel such as diesel fuel. The mixture ignites when the diesel fuel is injected.
As the directly injected diesel fuel is mixed with the low reactivity fuel, a gradient of
fuel reactivity is established throughout the cylinder. The combustion is staged, reducing
the rate of pressure rise, and progresses from regions of higher reactivity to regions of lower
reactivity. There are three regions of energy release with the RCCI technique. The initial
energy release is from the high reactivity diesel fuel, the second from a mixture of the two
fuels, and the third from the low reactivity fuel.
Low
reactivity
fuel
High
reactivity
fuel
Figure 7.23
Example RCCI dual fuel operation.
References
229
Since two fuels are used, the RCCI technique has a greater operating range than HCCI
and allows increased control of the start of combustion and the combustion duration. With
this technique, it is possible to optimize the fuel reactivity for a range of engine operating
conditions. For an engine with a 12:1 compression ratio, Bessonette et al. (2007) found
that the optimum cetane number was about 45, that is, diesel fuel, at low loads, and at
high loads, the optimum cetane number was about 27, that is, gasoline fuel. Therefore, as
the load increases, one could specify a change in the fuel mixture from 20% gasoline/80%
diesel at low load to 85% gasoline/15% diesel at high load.
Kokjohn et al. (2009) report net indicated thermal efficiencies of 50% (isfc = 170
g/kWh) for dual fuel low-temperature combustion in a heavy duty 2.44 L single-cylinder
diesel engine. They used port fuel injection of gasoline and direct injection of diesel fuel,
with about 80% of the fuel energy from gasoline at high load (11 bar) conditions. The
overall equivalence ratio was 𝜙 = 0.77 and a high level of 45% EGR was used. The
soot emissions were 0.004 g/kWh and NO𝑥 emissions were 0.01 g/kWh, both well below
emission standards, eliminating the need for exhaust after treatment. Similar results were
reported by Hockett and Marchese (2015) in a CFD study of RCCI.
As of 2014, a number of manufacturers have vehicle prototypes in development, with
planned commercial introduction in the coming decade. One approach that they have
chosen to meet the high load/speed issues is to use the engine in a dual mode, that is, lowtemperature combustion operation at lower loads, and spark or diesel operation at higher
loads.
For further information about combustion processes in engines, books by Borman and
Ragland (1998) and Turns (2012) cover combustion chemistry and kinetics in internal combustion engines from an engineering perspective. Detailed information about hydrocarbon
chemical kinetics is given in Westbrook and Dryer (1984) and in the text by Law (2006).
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NAJT, P. and D. FOSTER (1983), ‘‘Compression-Ignited Homogeneous Charge Compression,’’ SAE
Paper 830264.
RASSWEILER, G. and L. WITHROW (1938), ‘‘Motion Pictures of Engine Flames Correlated with Pressure
Cards,’’ A landmark reprint paper commemorating SAE’s 75th Anniversary, SAE paper 800131.
RICARDO, H. (1921), ‘‘The Influence of Various Fuels on the Performance of Internal Combustion
Engines,’’ Automot. Eng., Vol. 11, p. 92.
SIEBERS, D. (1998), ‘‘Liquid-Phase Fuel Penetration in Diesel Sprays,’’ SAE paper 980809.
Homework
231
SMITH, J., R. GREEN, C. WESTBROOK, and W. PITZ (1984), ‘‘An Experimental and Modeling Study of
Engine Knock,’’ Twentieth Symposium (International) on Combustion, Combustion Institute,
Pittsburgh, Pennsylvania.
SOID, S. and Z. ZAINAL (2011), ‘‘Spray and Combustion Characterization for Internal Combustion Engines Using Optical Measuring Techniques--A Review,’’ Energy, Vol. 36, No. 2, pp.
724--741.
TABACZYNSKI, R., F. TRINKER, and B. SHANNON (1980), ‘‘Further Refinement and Validation of a
Turbulent Flame Propagation Model for Spark Ignition Engines,’’ Combust. Flame, Vol. 39,
No. 2, p. 111--122.
TURNS, S. (2012), An Introduction to Combustion, 3rd Edition, McGraw-Hill, New York.
VAVRA, J., S. BOHAC, L. MANOFSKY, G. LAVOIE, and D. ASSANIS (2012), ‘‘Knock in Various Combustion Modes in a Gasoline Fueled Automotive Engine,’’ J. Eng. for Gas Turbines and Power,
Vol. 134, p. 082807-1 - 082807-8.
WESTBROOK, C. and F. DRYER (1984), ‘‘Chemical Kinetic Modeling of Hydrocarbon Combustion,’’
Prog. Energy Combust. Sci., Vol. 10, p. 157.
WITZE, P. and F. VILCHIS (1981), ‘‘Stroboscopic Laser Shadowgraph Study of the Effect of Swirl on
Homogeneous Combustion in a Spark Ignition Engine,’’ SAE paper 810226.
YOUNG, M. (1980), ‘‘Cyclic Dispersion-Some Quantitative Cause and Effect Relationships,’’ SAE
paper 800459.
7.7 HOMEWORK
7.1 Compare the laminar flame speed 𝑠l of isooctane, propane, and methanol for 𝑃 = 20 atm,
𝑇u = 600 K, 𝑓 = 0.05, and 𝜙 = 0.9, 1.0, 1.1, and 1.2. At what equivalence ratios are the
laminar flame speeds maximum for these fuels?
7.2 (a) Compute the laminar flame speed 𝑠l at ignition for an isooctane-fueled engine with a
compression ratio 𝑟 = 8 and spark timing 𝜃s = −25◦ atdc. The pressure and temperature at
the time of ignition are given by
𝑇u,s = 350(𝑉bdc ∕𝑉s )(𝛾−1)∕𝛾
𝑃s = 0.5(𝑉bdc ∕𝑉s )𝛾
The residual fraction is given by 𝑓 = 0.10 (8∕𝑟), the combustion is stoichiometric with
𝜙 = 1, and 𝛾 = 1.3.
(b) If the ignition delay Δ𝜃id is inversely proportional to the laminar flame speed at the
time of ignition, and Δ𝜃id = 25◦ for the conditions of part (a), plot the ignition delay Δ𝜃id
versus 𝜃s for −50◦ < 𝜃s < 0◦ and show lines of constant compression ratio for 𝑟 = 8 and
𝑟 = 10. Discuss the results.
7.3 A turbocharged diesel engine has a compression ratio of 16:1 and stroke of 150 mm, with
an inlet manifold conditions of 2 bar and 380 K. The cetane number of the diesel fuel is 40.
What is the ignition delay 𝜏id in crank angle degrees if 𝑁 = 2500 rpm? Assume a polytropic
coefficient 𝑛 = 1.35.
7.4 A maximum ignition delay 𝜏id of 20◦ is required for acceptable cold start ignition of an
automobile diesel engine. The engine stroke is 80 mm, the cranking speed 𝑁 is 200 rpm,
and the polytropic coefficient 𝑛 for compression to top dead center is 1.20. The fuel cetane
number is 45, and the inlet air temperature and pressure are 250 K and 1 bar. What is the
minimum compression ratio needed to have an ignition delay no longer than 20◦ ?
232
Combustion Processes in Engines
7.5 Consider the dependence of the precursor formation rate with temperature. For isooctane
at 𝑃 = 10 bar, plot the nondimensional precursor formation rate 𝑑𝜁 ∕𝑑𝑡 versus temperature.
Use the constants 𝐴 = 50.5, 𝐵 = 3800, and 𝑛 = 1.7, and vary the unburned gas temperature
𝑇u from 300 to 1100 K.
7.6 For an unburned gas temperature of 1200 K and pressure of 20 bar, with a combustion
duration of 60◦ , what is the critical engine speed 𝑁 above which knock will not have
enough time to occur? Use the constants 𝐴 = 50.5, 𝐵 = 3800, and 𝑛 = 1.7.
7.7 A combustion model produced the following data for an engine operated at wide open
throttle on isooctane, where 𝑇u is the temperature of the unburned gas mixture and 𝑃 is
the gas pressure. The start of combustion (𝑥b = 0) is −40◦ atdc, and the end of combustion
(𝑥b = 1.0) is +20◦ atdc:
𝜃 (◦ atdc)
𝑃 (bar)
−40
−35
−30
−25
−20
−15
−10
0
10
15
20
5.6
6.5
7.5
9.8
14.6
25.0
35.0
52.8
58.6
54.0
45.2
𝑇u (K)
600
620
650
690
745
820
880
965
996
990
975
If the precursor formation rate is given by Equation 7.10
𝑑𝜁
= 50.5 𝑃 1.7 exp (−3800∕𝑇u )
𝑑𝑡
a. Determine the minimum engine speed 𝜃̇ min for which knock-free operation occurs,
assuming the table is speed independent. Use the trapezoidal rule for integration
of Equation 7.15.
b. Plot the extent of reaction 𝜁 versus crank angle at that speed. Comment on assumptions implicit or explicit in the analysis.
c. Plot the extent of reaction 𝜁 versus crank angle at engine speeds of 70 and 130%
of the minimum engine speed 𝜃̇ min . For the slower engine speed, what is the crank
angle 𝜃 at which knock is predicted to occur?
7.8 Estimate the dominant acoustic frequency (kHz) in a knocking CFR engine if the average
end-gas temperature is assumed to be 1500 K.
7.9 The existence of a temperature gradient in the burned gas can be explained fairly simply
using an ideal gas model in which the fluid is broken into an ensemble of elements. The
average pressures and specific volumes of an Otto cycle are represented in Figure 7.24 by
the diagram 1-2-3-4. All the gas is compressed isentropically from 1 to 2; hence at point
2, the gas is at a uniform temperature T2. The first element (infinitesimal) to burn will not
Homework
Pressure
2"
233
3" 3 3'
2'
2
4''
4 4'
1
Figure 7.24 Ensemble of fluid
elements in Otto cycle for Problem 7.9.
Specific volume
influence the cylinder pressure, and thus burns at constant pressure to 2′ . Thus
𝑞
𝑇2′ = 𝑇2 +
𝑐p
where 𝑞 is the energy release per unit mass. That gas is then compressed isentropically to
the peak pressure 𝑃3 , hence
( )(𝛾−1)∕𝛾
𝑇3′
𝑃3
=
𝑃2
𝑇2′
The last element to burn is compressed isentropically as unburned gas to the peak pressure
at 2′′ .
( )(𝛾−1)∕𝛾
𝑇2′′
𝑃3
=
𝑇2
𝑃2
The last element then also burns at constant pressure so that
𝑞
𝑇3′′ = 𝑇2′′ +
𝑐p
All the elements expand isentropically after the last element burns.
Taking as the average cycle the conditions used in Figure 2.1 of Chapter 2, find
(a) The ratio 𝑇3′ ∕𝑇3′′
(b) The ratio 𝑣3′ ∕𝑣3′′
7.10 Derive Equation 7.26 for the diesel fuel injection rate. Assume 𝑢 = 𝑢(𝑇 , 𝜙).
7.11 An HCCI engine needs to compress a fuel--air mixture to 1300 K at top dead center for
proper autoignition. Assuming a residual gas fraction 𝑓 of 0.35 at a temperature of 750 K,
what should the inlet air temperature be if (a) the compression ratio is 11:1, and (b) the
compression ratio is 18:1? Assume the specific heat ratio 𝛾 = 1.35.
Chapter
8
Emissions
8.1 INTRODUCTION
In this chapter, we discuss how pollutants are formed during the combustion process in
an engine and examine measures that have been taken to reduce airborne emissions from
engines. The major emissions from internal combustion engines include nitrogen oxides
(NO𝑥 ), carbon monoxide (CO), hydrocarbons (HC), particulates (PM), and aldehydes.
These combustion products are a significant source of air pollution, as internal combustion
engines are the source of about half of the NO𝑥 , CO, and HC pollutants in the atmosphere.
The emissions from engines have a number of adverse health and environmental effects,
as many research studies have shown a strong correlation between air pollution levels
and human health effects. The health effects include reduced lung function, cardiovascular
issues, coughs, asthma, and eye irritation.
Nitrogen oxides are formed during the combustion process, and in the atmosphere react
with water vapor and solar radiation to form nitric acid, a component of acid rain, and ground
level ozone, O3 , a component of smog. In addition to creating significant respiratory system
problems, both acid rain and smog damage forests, streams, and agricultural products. If
transported over a wide area by prevailing winds, nitrogen oxides can create regional air
quality issues.
Carbon monoxide is a product of rich combustion and reacts with oxygen and nitrogen
oxides in the exhaust stream and atmosphere to form smog. When inhaled, carbon monoxide
interferes with oxygen distribution throughout the circulatory system due to its high affinity
for hemoglobin, about 200 times than that of oxygen. Carbon monoxide poisoning is the
most common type of fatal air poisoning worldwide. The U.S. Environmental Protection
Agency (EPA) first set air quality standards for CO in 1971, specifying an 8-h primary
standard at 9 parts per million (ppm) and a 1-h primary standard at 35 ppm.
Hydrocarbon emissions result from release of unburned or partially combusted hydrocarbon fuels. Hydrocarbons also contribute to the chemical reactions that form ground
level ozone. Various hydrocarbon compounds can cause increased incidence of respiratory
problems and lung cancer. Finally, inhalation of particulates from engines causes increased
respiratory problems.
Carbon dioxide (CO2 ), a primary gaseous combustion product of internal combustion
engines, is also a greenhouse gas and is in the process of being regulated as well, due to
its increasing atmospheric concentration. The combustion of fossil fuels such as coal and
petroleum is the leading cause of increasing CO2 concentration in the atmosphere. About
60% of the CO2 produced by combustion remains in the atmosphere; the remainder is
removed from the atmosphere by plant photosynthesis and by diffusion into ocean water.
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
234
Nitrogen Oxides
235
Before the advent of the industrial revolution, the average atmospheric concentration of
CO2 was about 280 ppm. Long-term precision monitoring began around 1960, when the
average atmospheric concentration of CO2 was measured at 316 ppm. By 1970, it was
325 ppm, in 1990, the concentration had risen to 354 ppm, and in 2010, it was 387 ppm,
an increase of about 1--2 ppm per year.
8.2 NITROGEN OXIDES
Nitrogen oxides (NO𝑥 ) are formed throughout the combustion chamber during the combustion process due to the disassociation of N2 and O2 into their atomic states and subsequent
reactions with molecular oxygen and nitrogen. The effect of engine operational parameters
including equivalence ratio, spark timing, engine speed, and manifold pressure on nitrogen
oxide concentration has been the goal of a great deal of research. The reactions forming NO𝑥
are highly temperature dependent, so NO𝑥 emissions are relatively low during engine start
and warm-up, and then scale proportionally with the engine load. The total nitrogen oxide
concentration is measured with a chemiluminescence analyzer, as discussed later in the text.
Nitrogen oxides (NO𝑥 ) are composed of NO and NO2 . Many complex reaction pathways for NO𝑥 creation and decay have been formulated, and the corresponding rate parameters for the reactions have been established. In spark ignition engine exhaust, the dominant
component of NO𝑥 is nitric oxide, NO, with concentrations of the order of 1000 ppm, and
the concentration of nitrogen dioxide, NO2 , is of the order of 10 ppm, that is, about 1%.
In compression ignition engines, the concentration of NO2 can be higher, approaching
10--30% of the total NO𝑥 . In the atmosphere, nitric oxide will oxidize to nitrogen dioxide
and react with unburned hydrocarbons in the presence of sunlight to form smog.
There are three major chemical mechanisms that produce NO. These are the thermal
or Zeldovich mechanism, the prompt or Fenimore mechanism, and the combustion of fuelbound nitrogen. For internal combustion engines, the most significant is the Zeldovich
mechanism in which NO is formed in the high-temperature burned gases behind by the
flame front. The prompt mechanism occurs within the relatively thin combustion flame
front. Since the volume of the high-temperature burned gases is much larger than the
instantaneous volume of the flame front, the amount of NO formed from the prompt
mechanism is relatively small compared with that formed from the thermal or Zeldovich
mechanism. Fuel-bound NO is formed from nitrogen in the fuel. Fossil fuels typically
contain 0.5--2.0% nitrogen by weight, so during combustion of nitrogen-containing fuels,
hydrogen cyanide and ammonia are formed, which react with O and OH to form NO.
The following three chemical equations form the extended Zeldovich reaction mechanism (Miller and Bowman, 1989):
O + N2
N + O2
𝑘1
⇌
𝑘1𝑟
𝑘2
⇌
𝑘2𝑟
N + OH
𝑘3
⇌
𝑘3𝑟
NO + N
(8.1)
NO + O
(8.2)
NO + H
(8.3)
The first two reactions, Zeldovich (1946), were proposed by Yakov Zeldovich
(1914--1987), a Soviet physicist. They are chain-branching reactions, as two radical
species are formed from a reaction that consumes only one radical. The first reaction,
236
Emissions
Equation 8.1, is a nitrogen dissociation reaction triggered by an oxygen atom. This reaction
is slow and therefore rate limiting, as it is endothermic with activation energy of 75.0 kcal.
The second reaction, Equation 8.2, is very fast, as a nitrogen atom reacts exothermically
(+31.8 kcal) with an oxygen molecule to form nitric oxide and an oxygen atom. The third
reaction, Equation 8.3, is an exothermic (+49.4 kcal) reaction between a nitrogen atom and
a hydroxide radical that forms nitric oxide and a hydrogen atom. This third reaction was
proposed by Lavoie et al. (1970), and assumes partial equilibrium of the reaction:
(8.4)
O + OH ⇌ O2 + H
The prompt mechanism, Fenimore (1971), occurs in rich combustion conditions at the
flame zone. It is a reaction sequence initiated by reaction of hydrocarbon radicals with
molecular nitrogen, leading to intermediate molecules, such as hydrogen cyanide, HCN,
that then react to form NO, with concentrations of the order of 50 ppm.
The NO formed in the flame zone can be converted to NO2 through the following
reaction:
(8.5)
NO + HO2 ⇌ NO2 + OH
and converted back to NO through reactions with O and H, for example,
NO2 + O ⇌ NO + O2
(8.6)
The rate constants for the extended Zeldovich reaction mechanism are given in Equation 8.7 (Hanson and Salimian, 1984). These rate constants are relatively slow compared with typical I. C. engine combustion timescales. The rate constants have units of
cm3 /(mol s), the additional subscript 𝑟 on the rate constants denotes the reverse reaction
rate constant, and the temperature 𝑇 is in Kelvin.
𝑘1 = 1.8 × 1014 exp(−38, 370∕𝑇 )
𝑘1𝑟 = 3.8 × 1013 exp(−425∕𝑇 )
𝑘2 = 1.8 × 1010 𝑇 exp(−4, 680∕𝑇 )
𝑘2𝑟 = 3.8 × 109 𝑇 exp(−20, 820∕𝑇 )
(8.7)
𝑘3 = 7.1 × 1013 exp(−450∕𝑇 )
𝑘3𝑟 = 1.7 × 1014 exp(−24, 560∕𝑇 )
Following Heywood (1976), one can write the following expression for the rate of
change of nitric oxide concentration, with the brackets denoting molar concentrations in
units of mol/cm3 .
𝑑
[NO] = +𝑘1 [O][N2 ] − 𝑘1𝑟 [NO][N] + 𝑘2 [N][O2 ]
𝑑𝑡
(8.8)
−𝑘2𝑟 [NO][O] + 𝑘3 [N][OH] − 𝑘3𝑟 [NO][H]
To apply Equation 8.8, two approximations are introduced. First, that the C-O-H
system is in equilibrium and is not perturbed by N2 dissociation, and second, the N atoms
change concentration by a quasi-steady process. The first approximation means simply that
given the pressure, temperature, equivalence ratio, and residual fraction of a fluid element,
one simply computes the equilibrium composition to determine the concentrations of N2 ,
O2 , O, OH, and H. The second approximation means that one can solve for the N atom
concentration by setting the rate of change of N atoms to zero:
𝑑
[N] = +𝑘1 [O][N2 ] − 𝑘1𝑟 [N][NO] − 𝑘2 [N][O2 ]
𝑑𝑡
+𝑘2𝑟 [NO][O] − 𝑘3 [N][OH] + 𝑘3𝑟 [NO][H]
=0
(8.9)
Nitrogen Oxides
237
With these two approximations, it can be shown that
2𝑅1 (1 − 𝛼 2 )
𝑑[NO]
=
𝑑𝑡
1 + 𝛼𝑅1 ∕(𝑅2 + 𝑅3 )
(8.10)
where 𝛼 is the ratio of the nitric oxide concentration to its equilibrium value:
𝛼=
[NO]
[NO]𝑒
(8.11)
and 𝑅𝑖 (𝑖 = 1, 2, 3) are various rates of reaction, with the equilibrium concentrations labeled
with the subscript 𝑒:
𝑅1 = 𝑘1 [O]𝑒 [N2 ]𝑒
𝑅2 = 𝑘2𝑟 [NO]𝑒 [O]𝑒
𝑅3 = 𝑘3𝑟 [NO]𝑒 [H]𝑒
The total amount of nitric oxide that appears in the exhaust is computed by summing
the mass fractions for all the fluid elements:
𝑥̄ NO =
∫0
1
(8.12)
𝑥NO 𝑑𝑥
The above NO reaction mechanism has been incorporated into the two-zone finite heat
release program Homogeneous.m introduced in Chapter 4. The program computes and
plots both the equilibrium and the rate-limited NO concentrations as a function of crank
angle, and also calculates NO concentration in the exhaust. Use of the program to compute
NO formation is detailed in the following example:
EXAMPLE 8.1
NO Formation with a Homogeneous Two-Zone Finite Energy Release Model
A CFR engine is operated with gasoline using the following baseline conditions given in
the table below. The start of heat release is −15◦ atdc and the burn duration is 45◦ . The
inlet conditions are 𝑃1 = 100 kPa and 𝑇1 = 350 K. Using the Homogeneous.m program,
plot the burn fraction, pressure, burned and unburned temperatures, equilibrium, and ratelimited NO values as a function of crank angle. This engine is similar to the CFR engine
used in the NO study by Komiyama and Heywood (1973).
Parameter
Value
Bore (m)
Stroke (m)
Half stroke/rod ratio
Compression ratio
Engine speed (rpm)
Equivalence ratio
Residual fraction
Cylinder heat transfer coefficient (W/(m2 K))
Mass blowby coefficient (s−1 )
Cylinder wall temperature (K)
0.0825
0.1143
0.25
7
1200
0.88
0.05
500
0.8
400
238
Emissions
SOLUTION The input parameters are entered into the Homogeneous.m program as shown
below:
% Homogeneous Two Zone Combustion Cycle
% This program computes the pressure and temperature
% vs crank angle, the work, indicated thermal efficiency
% and the Indicated mean effective pressure (kPa)
R = 10;
% Compression ratio
B = .10;
% Bore - B (m)
S = .08;
% Stroke - S (m)
EPS = 0.25;
% Half stroke to rod ratio
RPM = 2000;
% Engine speed (RPM)
HEAT = 500;
% Heat transfer coefficient (W/m2-K)
BLOWBY = 0.8;
% Blowby coefficient
THETAB = 60;
% Burn angle (Deg)
THETAS = -35;
% Start of heat release (deg ATDC)
PHI = 0.8;
% Equivalence ratio
F = 0.1;
% Residual fraction
TW = 420;
% Wall temperature (K)
fuel_id = 2;
FS = 0.06548;
% gasoline
% gasoline stoichiometric fuel-air ratio
A0 = 47870;
% maximum available energy (kJ/kg)
T1 = 350;
% Initial temperature (K)
P1 = 100;
% Initial Pressure (kPa)
...
The results of the computations are presented in the following four figures. The burn
fraction profile is plotted in Figure 8.1, the calculated pressure profile in Figure 8.2, and
the calculated unburned and burned temperature profiles in Figure 8.3. Each mass burns
at its adiabatic flame temperature based on the unburned gas temperature at the time it
burned. Once burned, the temperature of a mass tracks the pressure, as it is more or less
isentropically compressed or expanded. Notice that the first mass to burn is compressed
significantly. Each subsequent element to burn is compressed less, and the last element to
burn undergoes no compression. As a result, the first mass to burn is hotter than all the rest,
and the last mass to burn is the coolest.
The resulting NO profiles are plotted in Figure 8.4. Figure 8.4 illustrates how the nitric
oxides vary with time in different fluid masses as the burned volume increases. The equilibrium concentration is computed based on the local temperature, pressure, equivalence
ratio, and residual mass fraction.
The plot illustrates some important points about NO combustion. First, there is a
significant difference between the equilibrium and the rate-limited concentrations of NO
during the combustion process, as the rate-limited NO concentrations lag behind the equilibrium NO concentrations. Secondly, since the chemical reaction rates increase strongly
Nitrogen Oxides
239
0
Crank angle
50
100
0
Crank angle
50
100
1
0.8
0.6
0.4
0.2
Figure 8.1 Mass fraction
burned versus crank angle
(Example 8.1).
0
–100
–50
4000
Pressure (kPa)
3500
3000
2500
2000
1500
1000
500
0
–100
Figure 8.2 Pressure versus
crank angle (Example 8.1).
–50
3000
Unburned
Burned
Temperature (K)
2500
2000
1500
1000
500
Figure 8.3 Calculated
temperature of burned
gas 𝑇b and unburned gas
𝑇𝑢 (Example 8.1).
0
–100
–50
0
Crank angle
50
100
240
Emissions
Equilibrium NO
Figure 8.4 Predicted
equilibrium and rate-limited NO
concentrations (Example 8.1).
Crank angle
with temperature, there are large differences between the nitric oxide concentrations in the
first mass to burn (𝑥 = 0) and last mass to burn (𝑥 = 1).
Furthermore, it can be seen that when the temperatures drop to about 2000 K, the
decomposition rate becomes very slow, and for practical purposes, it may be said that the
nitric oxides freeze at a concentration greater than the equilibrium values. The relatively
high values of the ‘‘frozen’’ concentrations of nitric oxides in the exhaust are thus a function
of the gas temperatures during combustion, not temperatures in the exhaust.
Some additional computational results using the engine of Example 8.1 are now
presented to illustrate how nitric oxides in the exhaust depend on various engine parameters
such as equivalence ratio, residual fraction, spark timing, engine speed, and cylinder wall
temperature. The engine was operated at the baseline conditions given, and the various
engine operational parameters were individually varied.
Figure 8.5 shows a result typical of all engines that nitric oxides are maximized with
mixtures slightly lean of stoichiometric. Recall that increased temperatures favor nitric
oxide formation and that burned gas temperatures are maximized with mixtures that are
slightly rich. On the other hand, there is little excess oxygen in rich mixtures to dissociate
and attach to nitrogen atoms to form nitric oxide. The interplay between these two effects
results in maximum nitric oxides occurring in slightly lean mixtures, where there is a slight
excess of oxygen atoms to react with the nitrogen atoms.
As the residual fraction is increased, the NO levels decrease, since the effective specific
heat of the combusting mixture is increased. Note that since the program uses FARG to
compute the residual gas mixture concentrations, the NO concentration in the residual
fraction is not included in the exhaust NO concentrations.
Figures 8.6--8.10 lead to four additional observations, as follows:
• Increased cylinder wall temperature increases the nitric oxides, as the cylinder heat loss
is decreased.
• The dependence on spark timing is strong for lean mixtures. As the timing is advanced,
the NO levels increase, since the combustion temperatures increase.
• The dependence on engine speed has two competing factors to consider. These are the
time available for heat loss and the time available for combustion. As the engine speed
Nitrogen Oxides
Figure 8.5 Calculated
exhaust NO concentration
versus equivalence
ratio and residual fraction
(Example 8.1).
241
Equivalence ratio
6500
6000
5500
5000
4500
Figure 8.6 NO concentration
versus cylinder wall temperature
(Example 8.1).
4000
350
400
450
Cylinder wall temperature (K)
500
increases, the heat loss decreases, causing an increase in combustion temperature, and the
time available for combustion decreases, causing a decrease in combustion temperature.
The overall result can be a maximum NO level at an intermediate engine speed.
• Increasing the inlet manifold pressure, and thus the imep, will increase the NO concentration.
At this point, it is useful to discuss the mixing of the burned gases. Fluid elements
are mixed with one another via turbulence. If the rate of mixing is faster than the rate at
which burned gas is produced, then the burned gas can be assumed to be homogeneous
and characterized by a single temperature. If the mixing is slow, then the burned gas must
be treated as an ensemble of fluid elements at different temperatures. Experimentally, it
is observed that there are different temperature fluid elements in the burned gases but the
differences are smaller than predicted. Thus, it can be inferred that mixing occurs, but it is
not complete during combustion.
It can be shown using the analyses in Chapter 4 that the energy of the burned gas is a
nearly linear function of temperature (i.e., the specific heat variation is relatively small over
the range of temperatures encountered in the burned gas) so that for computing cylinder
pressure, the overall average gas temperature can be used. The same cannot be said of
242
Emissions
8000
7000
6000
5000
4000
Figure 8.7 NO concentration
versus start of heat release
(Example 8.1).
3000
–30
–25
–20
–15
–10
Start of heat release (Deg)
5000
4500
4000
3500
Figure 8.8 NO concentration
versus engine speed (Example 8.1).
3000
500
1000
1500
2000
2500
3000
Engine speed (RPM)
7000
6000
5000
4000
3000
Figure 8.9 NO concentration
versus IMEP (Example 8.1).
2000
400
600
800
1000
IMEP (kPa)
1200
1400
Carbon Monoxide
243
4500
Nitric oxide concentration (ppm)
N = 2500 rpm
N = 1500 rpm
4000
= 0.97
3500
= 1.3
3000
2500
2000
1500
1000
500
Figure 8.10 Advanced timing increases NO.
Adapted from Huls and Nickol (1967).
0
20
30
40
15 20 25 30
nitric oxides since the chemical rates are nonlinear functions of temperature. Using the
Farg/Ecp formulation detailed in Chapter 4, Raine et al. (1995) have computed the nitric
oxide formation in a multizone model. The NO model above does not include mixing and
the temperature gradients due to wall boundary layers. The state of the art requires one to
account for these effects to realize good agreement with experiment under all circumstances.
The trends shown above, although typical for homogeneous spark ignition engines,
are by no means universal, especially for compression ignition engines. With compression ignition engines, one has to further account for variations in equivalence ratio from
fluid element to fluid element. More complex combustion models also use detailed NO𝑥
mechanisms that include more than 50 reactions.
8.3 CARBON MONOXIDE
Carbon monoxide (CO) appears in the exhaust of rich-running engines, since there is
insufficient oxygen to convert all the carbon in the fuel to carbon dioxide. Another CO
source is the dissociation of CO2 at high combustion temperatures. The formation and
destruction of CO is a principal reaction pathway in hydrocarbon combustion, which
essentially consists of breakdown of the hydrocarbon fuel to carbon monoxide, and then
the oxidation of carbon monoxide to carbon dioxide. The majority of the heat release for
the combustion process occurs during the CO oxidation, given by the following reaction:
CO + OH ⇌ CO2 + H
(8.13)
The most important engine parameter influencing carbon monoxide levels is the fuel-air equivalence ratio. Thus, results obtained when varying the fuel--air ratio are more or less
universal. Typical results are shown in a classic plot of concentration versus fuel--air ratio,
Figure 8.11, for a supercharged engine fueled with C8 H18 . Notice that at near stoichiometric
conditions, carbon monoxide emission is a highly nonlinear function of equivalence ratio.
Under these circumstances, in multicylinder engines, it becomes important to ensure that
the same fuel--air ratio is delivered to each cylinder. If half the cylinders run lean and the
Emissions
14
12
CO2
10
Concentration (%)
244
8
Lean
Rich
6
CO
4
H2
O2
2
Figure 8.11 Exhaust gas
composition versus fuel--air ratio
for supercharged engine with
valve overlap; fuel C8 H18 (Gerrish
and Meem, 1943).
0
C4H8
CH4
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
Fuel–air ratio (oxidized exhaust)
other half run rich, then the lean cylinder produces much less CO than the rich cylinders. The
average CO emission of such an engine would correspond not to the average equivalence
ratio but to an equivalence ratio richer than average, producing more CO than is necessary.
The C-O-H system is more or less in equilibrium during combustion and expansion
up to the point where the nitric oxide chemistry freezes. Thus, whether it is a lean- or
rich-running engine, one can determine the carbon monoxide concentration during these
times using equilibrium chemistry assumptions. Late in the expansion stroke, with the gas
temperatures down to about 1800 K, the chemistry in C-O-H systems starts to become
controlled by chemical kinetics and is generally frozen by the time blowdown finishes, and
the exhaust valve opens. Therefore, the measured values of CO in exhaust gases are lower
than the peak values in the combustion chamber, but greater than equilibrium conditions
for the exhaust gases.
The rate controlling reactions in the C-O-H systems are three-body recombination
reactions such as
H + H + M ⇌ H2 + M
(8.14)
H + OH + M ⇌ H2 O + M
(8.15)
H + O2 + M ⇌ HO2 + M
(8.16)
Results obtained by using an unmixed model for the burned gas and accounting for
these rate-limiting reactions are illustrated in Figure 8.12. In these plots, 𝑥 is the fraction of
the total charge burned when an element is burned and 𝑧 is the mass fraction that has left the
cylinder at the time an element leaves the cylinder. Because that time is unknown, results
are given for several values of 𝑧 for each element. Gas that leaves early (𝑧<<1) cools more
rapidly than gas that leaves last (𝑧 ≃ 1). The results show that gases that burn early carry
more CO into the exhaust than gases that burn later. They also show the fortuitous fact
that the frozen concentrations are close to the equilibrium concentrations that exist in the
Hydrocarbons
3
z = 0.01
z = 0.50
z = 0.99
z = Equilibrium CO
x = 0.05
COe
10–2
245
x = 0.50
CO mole fraction
COe
z = 0.01
COe
z = 0.50
10–3
Figure 8.12 CO concentration in
two elements of the charge that
burned at different times during
the combustion process; 𝑥 is the mass
fraction burned when the element
burned and 𝑧 is the fraction of gas
that has left the cylinder during the
exhaust process (Heywood, 1976).
z = 0.01
Exhaust
valve
opens
3 x10 –4
tdc
30
0
60
90 120
z = 0.99
o
bdc
atc
150 180 210
10
5
Time (ms)
cylinder at the time the exhaust valve opens. This suggests an approximation that is often
used in practice, to assume that the C-O-H system is in equilibrium until the exhaust valve
opens, at which time it then freezes instantaneously. In lean-running engines, there appears
to be an additional source of CO caused by the flame--fuel interaction with the walls, the oil
films, and the deposits. Under these circumstances, the exhaust concentrations are so low
that they are not a practical problem, and thus details of these interactions remain largely
unexplored.
Thus, the key to minimizing CO emissions is to minimize the times the engine must
run rich (such as during start-up). Since diesel engines run lean overall, their emissions of
carbon monoxide are low and generally not considered a problem. However, it does appear
that direct injection diesel engines emit relatively more CO than indirect injection diesel
engines.
8.4 HYDROCARBONS
Hydrocarbon emissions result from the presence of unburned fuel in the exhaust of an engine. Hydrocarbon emissions are greatest during engine start and warm-up, due to decreased
fuel vaporization and oxidation. Hydrocarbon emissions from spark ignition engines and
diesel engines are discussed separately in this section, as their combustion processes and
resulting hydrocarbon emission sources are different.
Hydrocarbon emissions from engines have been grouped into a number of classifications for regulatory purposes. Two general classifications that are widely used are total
hydrocarbons (THC) and nonmethane organic gases (NMOG). Methane is a greenhouse
gas, but less photochemically reactive than the other hydrocarbons, so it is given a separate
classification. The level of hydrocarbon emissions is usually expressed in terms of the total
hydrocarbon concentration in parts per million carbon atoms.
246
Emissions
Table 8.1 Hydrocarbon Emission Sources
Source
Crevices
Oil layers
Deposits
Liquid fuel
Flame quench
Exhaust valve leakage
Total
% HC Emissions
38
16
16
20
5
5
100
Source: Cheng et al., 1993.
Hydrocarbon fuels are composed of 10--20 major species and some 100--200 minor
species. Most of these same species are found in the exhaust. However, some of the exhaust
hydrocarbons are not found in the parent fuel but are hydrocarbons derived from the fuel
whose structure was altered within the cylinder by chemical reactions that did not go to
completion. These hydrocarbons represent about 50% of the total hydrocarbons emitted.
The partial reaction hydrocarbon products, including acetaldehyde, formaldehyde, 1,3butadiene, and benzene, are classified by the U.S. Environmental Protection Agency as
toxic emissions.
Spark Ignition Engines
About 9% of the fuel supplied to a spark ignition engine is not burned during the normal
combustion phase of the expansion stroke. There are additional pathways that consume
7% of the hydrocarbons during the other three strokes of the four-stroke spark ignition
engine so that about 2% of the remaining unburned fuel will go out with the exhaust
(Cheng et al., 1993). As a consequence, hydrocarbon emissions cause a decrease in engine
thermal efficiency, as well as being an air pollutant. As listed in Table 8.1, six principal
mechanisms are believed to be responsible for the alternative oxidation pathways and the
exhaust hydrocarbons appearing: (1) crevices, (2) oil layers, (3) carbon deposits, (4) liquid
fuel, (5) cylinder wall flame quenching, and (6) exhaust valve leakage.
The crevice mechanism is the most significant, responsible for about 38% of the
hydrocarbon emissions. Crevices are narrow regions in the combustion chamber into which
a flame cannot propagate. When a flame tries to propagate into a narrow channel, it may or
may not be extinguished depending upon the ratio of the channel size and a characteristic
of fuel--air mixtures called the quenching distance. By definition, crevices in engines have
a characteristic size less than the quenching distance, so the cylinder gases in the crevices
will not be consumed during the cylinder combustion process. Crevices occur around the
piston, head gasket, spark plug, and valve seats, representing about 1--2% of the clearance
volume. The largest crevice is the piston ring--liner crevice region. During compression
and the early stages of combustion, the cylinder pressure rises, forcing a small fraction of
the fuel--air mixture into the crevices. The crevice temperatures are approximately equal
to the cooled wall temperatures, so the density of the fuel--air mixture in the crevices is
greater than in the cylinder. When the cylinder pressure decreases during the latter portion
of the expansion stroke, the unburned crevice gases will flow back into the cylinder, and
contribute to the hydrocarbon emissions.
Wentworth (1971) was one of the first to recognize the importance of the crevice
volume around the piston. He designed a special ring package to eliminate it. So far
Hydrocarbons
247
it has found application only in research engines, where it allows one to make crevice
hydrocarbons a negligible source. This allows study of the effects of engine variables on
the remaining sources.
Oil layers within an engine can also trap some of the fuel and later release it during
expansion. Kaiser et al. (1982) added oil to the engine cylinder and found that the exhaust
hydrocarbons increased in proportion to the amount of oil added when the engine was
fueled on isooctane. They verified that the increased emissions were unburned fuel and
fuel oxidation species and not unburned oil and oil oxidation species. They also did experiments in which the engine was fueled with propane and found no increase in the exhaust
hydrocarbons when oil was added. Since propane is not soluble in the oil, they concluded
that the increase observed is caused by fuel having been absorbed into the oil layer during
compression later being released into the cooling burned gas during the expansion stroke.
Thus, one can conclude that hydrocarbon emissions from engines will also depend on the
amount of oil in the cylinder and the solubility of the fuel in the oil.
With continued use, carbon deposits build up on the valves, cylinder, and piston heads
of internal combustion engines. The deposits are porous, and the sizes of the pores in the
deposits are smaller than the quenching distance, and as a result the flame cannot burn
the fuel--air--residual gas mixture compressed into the pores. This mixture comes out of
the pores during expansion and blowdown. Although some of it will burn up when mixed
with the hotter gases within the cylinder, eventually cylinder gas temperatures will have
dropped to the point where those reactions fail to complete, resulting in hydrocarbons being
emitted from the engine.
Fuel injection past an open valve into the cylinder, as can be the case with port fuel
injection, allows the fuel to enter the cylinder in the form of liquid droplets. The less
volatile fuel constituents may not vaporize, especially during engine start and warm-up,
and be adsorbed in the crevices, oil layers, and carbon deposits.
Flame quenching along the surfaces is a relatively minor mechanism. Daniel (1957)
showed that as the flame propagates toward the walls in an engine, it is extinguished at a
small but finite distance away from the wall. Flame photographs revealed a dark region near
the wall of thickness about one-half the quench distance. Measurements have shown that
these hydrocarbons are subsequently oxidized with a high efficiency as they diffuse into
the burned gases during expansion, and thus do not contribute significantly to the engine
out hydrocarbon emissions.
Hydrocarbons are exhausted from the engine in a complex process. At the end of
combustion, there are hydrocarbons all along the walls trapped in deposits, oil layers, or
the crevice volume. During expansion, hydrocarbons leave the crevice volume and are
distributed along the cylinder wall. When the exhaust valve opens, the large rush of gas
escaping drags with it some of the hydrocarbons released from oil layers and deposits.
During the exhaust stroke, the piston rolls the crevice volume hydrocarbons that were
distributed along the walls into a vortex that ultimately becomes large enough that a
portion of it is exhausted.
As shown schematically in Figure 8.13, Tabaczynski et al. (1972) have verified in a
water analog experiment that the piston rolls the wall layer into a vortex. They have also
measured the hydrocarbon emission mass flow rate as a function of time during the exhaust
stroke, plotted in Figure 8.14. Their results show that roughly one-half the hydrocarbons in
their engine are exhausted during blowdown and one-half are exhausted during the latter
portion of the exhaust stroke. The concentration profile has a peak during blowdown and a
sudden increase at about 290◦ , when the vortex starts to exit the combustion chamber. Twostroke spark ignition engines can produce a significant amount of hydrocarbon emissions.
Short-circuiting of the fuel--air mixture during the scavenging process is the major source
248
Emissions
Exhaust gas
Roll vortex
Piston
Figure 8.13 Wall vortex formed by exhaust stroke. Adapted
from Tabaczynski et al. (1972).
Figure 8.14 Variation of HC
concentration at the exhaust
valve during the exhaust
process. Adapted from
Tabaczynski et al. (1972).
Hydrocarbon concentration at
exhaust valve (ppm n-hexane)
800
700
600
500
400
300
200
100
0
100 120 140 160 180 200 220 240 260 280 300 320 340
Crank angle (deg)
of the hydrocarbon emissions. Direct fuel injection is increasingly being used in twostroke engines to eliminate the short-circuiting of the fuel during scavenging. If crankcase
compression is used, the unburnt lubrication oil is also a source of hydrocarbon emissions.
The hydrocarbon story does not end once the hydrocarbons leave the cylinder. There
is considerable burn up in the exhaust port. Some emission control techniques use an air
pump to inject ambient air into the exhaust manifold to further oxidize the hydrocarbons.
Compression Ignition Engines
Hydrocarbons from diesel engines come primarily from (1) fuel trapped in the injector at
the end of injection that later diffuses out, (2) fuel mixed into air surrounding the burning
spray so lean that it cannot burn, and (3) fuel trapped along the walls by crevices, deposits,
or oil due to impingement by the spray (Greeves et al., 1977 and Yu et al., 1980).
The diesel combustion process relies on mixing fuel and air at the time they are intended
to burn. As already mentioned, a combustion delay time is required for enough precursors
to form in order for autoignition of the fuel to occur. The delay time is a strong function
of equivalence ratio and is a minimum near stoichiometric proportions of fuel and air.
However, since the delay time is finite, a fuel--air pocket can be mixed to stoichiometric
249
Particulates
2000
HC (ppmc)
1500
1000
DI
500
IDI
Figure 8.15 HC concentrations as a function of
load for direct injection and prechamber engines.
Adapted from Pischinger and Cartellieri (1972).
0
0
1
2
3
4
5
bmep (bar)
6
7
proportions and then diluted by more air before autoignition occurs in that element. As a
result, there are contours for lean equivalence ratios, and when autoignition occurs, there
are fuel and air mixed locally to proportions less than the lean flammability limit. Thus,
this local fuel mixture does not burn and will increase the hydrocarbon emission levels.
There is also fuel mixed too rich to burn at the time of autoignition. However, it will
burn later with additional mixing, provided the gases are hot enough. Some hydrocarbons
are also produced because some of this fuel does not have a stoichiometric air--fuel ratio to
burn until late in the expansion stroke.
Pischinger and Cartellieri (1972) measured hydrocarbon emissions from both a naturally aspirated direct injection engine and a naturally aspirated indirect injection engine, see
Figure 8.15. The results, although unique to the engines in question, indicated that in general, direct injection engines emit more hydrocarbons than indirect injection engines. For
the direct injection engine, the hydrocarbons are greatest at light load. Thus, hydrocarbon
emissions at idle have been a focus of attention. Pischinger and Cartellieri (1972) found that
they can be influenced considerably by rather small changes in engine or injector geometry.
Nozzles manufactured by different companies from the same specifications produced very
different HC emissions. This illustrates the need for precise control in manufacturing to
achieve low emissions. The other variable examined was the piston bowl diameter, which
had a negligible influence on the bsfc, caused a slight change in NO𝑥 , and a dramatic
change in the hydrocarbons.
8.5 PARTICULATES
A high concentration of particulate matter (PM) is manifested as visible smoke or soot
in the exhaust gases. Particulate emissions from engines are regulated, since inhalation of
small particulate matter can create respiratory problems. Particulates are a major emissions
problem for diesel engines, as their performance is smoke limited. With the mandated
use of unleaded fuel for spark ignition engines, particulates are currently not as serious a
problem for spark ignition engines as they are for diesel engines.
The U.S. Environmental Protection Agency defines a particulate as any substance
other than water that can be collected by filtering diluted exhaust at or below 325 K. The
250
Emissions
particulate material collected on a filter is generally classified into two components. One
component is a solid carbon material or soot, and the other component is an organic fraction
consisting of hydrocarbons and their partial oxidation products that have been condensed
onto the filter or adsorbed to the soot. The organic fraction is influenced by the processes
that dilute the exhaust with air upon expulsion from the engine. The methods used to
measure particulate emissions are dilution tunnels, light absorption, filter discoloration,
and filter paper trapped mass, and are discussed in more detail in Chapter 12.
Inspection of the soot fraction under an electron microscope reveals it to be agglomerates of spherical soot particles approximately 200 Å in diameter. The agglomerates can
resemble a bunch of grapes in a more or less spherical configuration or be branched and
chainlike in character. The agglomerates pose a health hazard because they are too small
to be trapped by the nose and large enough that some particle deposition in the lungs
occurs. Because of their small size, particles on the order of ∼10 microns or less (PM10 )
can penetrate the deepest part of the lungs such as the bronchioles or alveoli. Particles less
than 2.5 microns in diameter (PM2.5 ) are referred to as ‘‘fine’’ particles and are believed
to pose the largest health risks. Particles with diameters between 2.5 and 10 microns are
referred to as ‘‘coarse.’’
The organic fraction results from all the processes that generate hydrocarbons and
their partial oxidation products. During the dilution process, some of them cool enough to
condense or adsorb the soot. In addition, some species originating from the lubricating oil
are found in the particulate and may be anywhere from 25 to 75% of the organic fraction
(Mayer et al., 1980).
Smoke forms in diesel engines because diesel combustion is heterogeneous. Even
though the diesel engine combustion process is lean overall, there are regions of fuelrich combustion, in the premixed and in the mixing-controlled combustion phases of
combustion, as shown by the combustion diagnostics presented in Chapter 7.
Consider a simple soot formation model with a two-stage reaction path for the
fuel-rich combustion of a hydrocarbon fuel:
First stage: formation of CO
b
Ca Hb + c O2 ⟶ 2c CO + H2 + (a − 2c)C(s)
2
Second stage: oxidation of CO
1
CO + O2 ⟶ CO2
2
C(s) + O2 ⟶ CO2
1
H2 + O2 ⟶ H2 O
2
According to this model, combustion takes place in two stages. If in the first stage,
𝑐 < 𝑎∕2, there is not enough oxygen present to convert all the carbon in the fuel to carbon
monoxide, the carbon--oxygen ratio >1, resulting in the production of soot or solid carbon
C(s). This is likely to occur locally within the fuel spray injected into the engine, since it
takes time for air and its attendant oxygen to be mixed in with the fuel. The sooting starts
with the formation of soot precursor species such as polycyclic aromatic hydrocarbon
(PAH) molecules, then the growth and agglomeration of these molecules into particles.
The second stage burns the CO, soot, and other first stage products to completion in
a diffusion flame. If there is enough oxygen present, that is, 𝑐 ≥ 𝑎∕2, then the flame is
Emissions Regulation and Control
251
–3
10
= 0.7
= 0.5
= 0.3
Particulate mass fraction
1500 rpm
Tfilter = 325 K
Figure 8.16 Particulate
emission versus dilution ratio
and equivalence ratio (Gillette
and Ferguson, 1983).
–4
10
10–5
0
10
20
30
40
50
Dilution ratio
clean since no solid carbon is formed. Incomplete oxidation will result in a sooting flame.
A measure of the sooting tendency of a fuel is termed the smoke point.
The detailed chemical processes leading to the formation of soot remain an active
area of research. Measurements indicate that diesel soot particles contain hydrogen as
well as carbon, with a C/H ratio of about 8. Particulate measurements by Gillette and
Ferguson (1983) obtained using a direct injection diesel with a dilution tunnel are shown in
Figure 8.16.
The amount of particulates is extremely dependent on the equivalence ratio. As the
equivalence ratio is doubled, the particulates increase by an order of magnitude. Wood
et al. (1982) measured the mass spectra of the organic fraction for particulates at three
different engine speeds, 𝜙 = 0.5, and a dilution ratio of 30. They reported that the spectrum
changed with engine speed, particularly for molecular weights greater than 300. The mean
molecular weight of the organic fraction was on the order of 200.
It is challenging to diesel engine designers that, generally, when a reduction in nitric
oxides has been achieved, it is at the expense of an increase in soot. Using the diesel
combustion model of Chapter 7, this is due to the fact that a decrease in the temperature
of the diffusion flame will not only decrease the NO𝑥 formation, but also decrease the
oxidization of soot.
Figure 8.17 is a representative plot indicating that as the timing is retarded, the NO𝑥
decreases, but the particulates increase, creating a trade-off between NO𝑥 and smoke.
Techniques used to decrease soot are to advance the injection timing and decrease the
amount of EGR (see Figure 8.18). Another technique used to decrease soot is to increase
the in-cylinder turbulence during the late stages of combustion. This increase in turbulence
can be accomplished through the use of auxiliary gas injection (Kurtz et al., 2000).
8.6 EMISSIONS REGULATION AND CONTROL
Prior to the mid-1960s, exhaust emissions from vehicles were uncontrolled. Since then
exhaust emissions from engines have been regulated worldwide by governmental agencies.
Emissions
–4
Soot (g/kWh)
0.2
–8
0.1
–12
–15
–18
0
Figure 8.17 Example plot of
soot and NO𝑥 trade-off
versus injection timing.
0
4
6
8
10
12
NOx (g/kWh)
0.2
Soot (g/kWh)
252
40
30
0.1
20
EGR (%)
10
Figure 8.18 Example plot of
soot and NO𝑥 trade-off
versus EGR.
0
0
4
6
8
10
12
NOx (g/kWh)
In the United States, exhaust regulations are set by the U.S. Environmental Protection
Agency. In 1966, in response to statewide air quality problems, The state of California
introduced hydrocarbon and CO emission limits for vehicles. In 1968, the United States
adopted the Clean Air Act that regulated vehicular and stationary emissions on a nationwide
basis. The 1968 Clean Air Act requires the EPA to set national ambient air quality standards
for ‘‘criteria pollutants.’’ Currently, there are six ‘‘criteria pollutants’’:
• carbon monoxide,
• nitrogen oxides,
• sulfur oxides,
• ozone,
• lead, and
• particulate matter.
During the intervening years, emission requirements have become increasingly rigorous, and internal combustion engines today are allowed to generate significantly less
pollution than their 1968 counterparts. Meeting these emission requirements has been a
major challenge and also an opportunity for automotive engineers. As shown in Figure 8.19,
Emissions Regulation and Control
253
Under floor
catalytic converter
Engine control
computer (ECU)
Tail pipe
Sensors
PCV valve
Figure 8.19 Engine
emission control methods
(Courtesy Englehard
Corporation).
Evaporative
emissions system
Close coupled
catalytic converter
EGR valve
Engine modifications
High energy
ignition
Low thermal inertia
Pipes and manifold
there are two basic methods used to control engine emissions: control of the combustion
process, and the use of after-treatment devices in the exhaust system.
The current and past emission regulations for vehicles are tabulated in Table 8.2,
and the emission standards for certification as a low emission vehicle (LEV) or ultralow
emission vehicle (ULEV) are given in Table 8.3. Note that the emission requirements for
nonstationary sources (vehicles) have units of grams per mile. The current hydrocarbon
emission limits have been reduced to 0.9%, nitrogen oxides to 1.7%, and carbon monoxides
to 4%, respectively, of the uncontrolled pre-1968 values.
Internal combustion engines used in applications other than vehicles, for example,
large engines used in locomotive and marine applications, stationary power plants, and
small engines used in nonroad applications, for example, lawn mowers, snow blowers,
chainsaws, pumps, and generators, are also now regulated, since they also have been found
Table 8.2 U.S. Passenger Car and Light-Duty Truck Emission Standards (g/mile)
Year
HC
Precontrol
1968
1972
1975
1977
1980
1981
1993
1994
2009
10.60
4.10
3.00
1.50
1.50
0.41
0.41
0.25
0.25
0.10
NO𝑥
CO
4.1
84.0
34.0
28.0
15.0
15.0
7.0
3.4
3.4
3.4
3.4
3.1
3.1
2.0
2.0
1.0
1.0
0.4
0.07
Table 8.3 Low Emission Vehicle (LEV) and Ultralow Emission Vehicle (ULEV) Standards
(g/mile)
LEV
ULEV
𝑎 Nonmethane
organic gas.
NMOG𝑎
NO𝑥
CO
0.075
0.040
0.2
0.2
3.4
1.7
254
Emissions
to be significant sources of hydrocarbon and carbon monoxide pollution. Stationary sources
are regulated in terms of emissions normalized by the energy output, that is, g/kWh.
Combustion Process Control
Application of technological advances in fuel injectors, oxygen sensors, and onboard
computers to engines has increased the control and subsequent optimization of the
engine combustion process. Combustion process improvements include increased fluid
turbulence and fuel mixing in the cylinder. As discussed earlier in the text, these improvements include modification of the intake valve size and position, and use of direct fuel
injection into the cylinder. The use of alternative and oxygenated fuels to reduce emissions
is the subject of Chapter 9.
Ignition Timing and Exhaust Gas Recirculation
Two NO𝑥 control measures that have been used in automobile engines since the 1970s
are retard of the ignition spark and exhaust gas recirculation (EGR). The aim of these
measures is to reduce the peak combustion temperature and thus the formation of NO𝑥 . As
shown in Figure 8.7, retarding the spark timing lowers the NO𝑥 , since a greater fraction of
the combustion occurs in an expanding volume, lowering the peak cylinder pressure and
temperature. However, this also decreases the engine thermal efficiency.
With the use of exhaust gas recirculation, some fraction of the exhaust gas is routed
back into the intake manifold. The nonreactive exhaust gas acts as a diluent in the fuel--air
mixture, lowering the combustion temperature, and the NO𝑥 formation. This is suggested
by the dependence on residual fraction plotted in Figure 8.5. The dilution by EGR of
the mixture also reduces the combustion rate, so the spark timing is usually advanced to
maintain optimal thermal efficiency. The EGR fraction increases with engine load up to
the lean limit, which is about 15--20% of the fuel--air flow rate.
Catalytic Converters
A variety of exhaust after-treatment devices have been added to vehicles to meet emission
requirements. These include catalytic converters, NO𝑥 traps, and particulate filters. Currently, the most important after-treatment device is the three-way heterogenous catalyst
(Kummer, 1981), invented in 1950 by Eugene Houdry, and first installed on the exhaust
systems in passenger cars in 1975. It derives its name from the fact that it works on all
three of the gaseous pollutants of concern: carbon monoxide, hydrocarbons, and nitric oxides. Heterogeneous catalysts provide a solid surface for gaseous reactions and reduce the
activation energy required for the reaction. When the reactants are adsorbed on the catalyst
surface, their internal molecular bonds are weakened, new bonds are more easily formed
between fragments of different molecular species, forming reaction products, which then
desorb from the catalyst surface back into the exhaust stream.
The surface area for reaction is important as it determines the number of catalytic
sites. All catalytic converters are built in a porous honeycomb or pellet geometry (see
Figure 8.20) to expose the exhaust gases to a larger surface made of small particles
(<50 nm) of one or more of the noble metals, platinum (Pt), palladium (Pd), and rhodium
(Rh). Platinum is the principal metal used to remove HC and CO, and rhodium is the
principal metal used to remove NO. Figure 8.21 is a schematic of a three-way honeycomb
catalyst. In the converter shown, a thin layer of the noble metals covers a washcoat of inert
Emissions Regulation and Control
255
Figure 8.20 Catalytic
converter (Courtesy
Englehard Corporation).
Ceramic
honeycomb
catalyst
Insulation
Insulation cover
Outlet
Inlet
Shield
Figure 8.21 Catalytic
converter components
(Courtesy Englehard
Corporation).
Coating
(alumina)
+ Pt/Pd/Rh
O, NO
HC, C
Catalyst
halfshell
housing
Intumescent mat
Substrate
alumina Al2 O3 on a cordierite honeycomb foundation. The operation of the catalytic converter is severely inhibited by lead and sulfur compounds in the exhaust so that vehicular
fuels have been reformulated to reduce their lead and sulfur content.
The reactions removing CO and HC are oxidation reactions, as shown in Equations
8.17 and 8.18, thus work best in a lean environment with 𝜙 < 1 , as the excess oxygen will
help drive the reactions to completion.
2CO + O2 ⟶ 2CO2
C𝑥 H2𝑥+2 +
3𝑥 + 1
O2 ⟶ 𝑥CO2 + (𝑥 + 1)H2 O
2
(8.17)
(8.18)
Emissions
100
Conversion efficiency (%)
256
Exhaust gas
80
CO
60
HC
40
20
Figure 8.22 Conversion
efficiencies for oxidizing
catalysts. Adapted from
Mondt (2000).
0
200
300
400
500
The reactions removing the NO are reduction reactions involving CO, H2 , and HC.
Since these are reduction reactions, they work best in a rich environment with 𝜙 > 1. Two
representative reactions are given in Equations 8.19 and 8.20,
1
(8.19)
N + CO2
2 2
1
(8.20)
NO + H2 ⟶ N2 + H2 O
2
As the exhaust gases flow through the catalyst, the CO and hydrocarbons are removed
through the above oxidation reactions forming CO2 and H2 O products. The oxidation rate
of hydrocarbons increases with molecular weight so that the oxidation of low molecular
weight fuels such as methane is very slow in the converter. The NO reacts with the CO,
hydrocarbons, and H2 via reduction reactions on the surface of the catalyst.
The catalytic conversion efficiency is plotted versus temperature in Figure 8.22. The
temperature at which a catalytic converter becomes 50% efficient is defined as the light-off
temperature. The light-off temperature is about 220◦ C for the oxidation of CO and 270◦ C
for the oxidation of HC. The conversion efficiency at fully warm conditions is about 98-99% for CO and 95% for HC, depending on the HC components. Various measures have
been tried to decrease the converter warm-up time, including use of an afterburner, locating
the converter or an additional start-up converter closer to the exhaust manifold, and electric
heating.
A three-way catalyst will function correctly only if the exhaust gas composition corresponds to nearly 1% stoichiometric combustion. If the exhaust is too lean, nitric oxides
are not destroyed and if it is too rich, carbon monoxide and hydrocarbons are not destroyed
(see Figure 8.23). Herein lies one constraint that emission control imposes upon engine
operation; to use a three-way catalyst, the engine must operate in a narrow window about
stoichiometric fuel--air ratios. As discussed in Chapter 12, a closed-loop control system
with an oxygen sensor is used to determine the actual fuel--air ratio, and adjust the fuel
injector so that the engine operates in a narrow range about the stoichiometric set point.
Ordinary carburetors run rich and are not able to maintain the fuel--air ratio in such a narrow
set point range.
Analysis of fuel--air cycles in Chapter 4 showed that lean operation was beneficial to the
thermal efficiency of the engine, and at first, it appears that operation at only stoichiometric
conditions is a rather severe constraint. However, if one realizes that the excess air in
NO + CO ⟶
Emissions Regulation and Control
257
100
90
NOx
Catalyst efficiency (%)
80
70
60
HC
CO
50
40
80%
efficiency
30
window
20
Stoichiometric
10
Figure 8.23 Conversion
efficiencies for three-way catalyst
versus air--fuel ratio (Kummer, 1981).
0
14.3
14.4
Rich
14.5
14.6
14.7
14.8
Lean
14.9
Air–fuel ratio
lean combustion is acting only as a dilutant, then one can appreciate that exhaust gas
recirculation can be used to achieve the same effect. The difference is that the excess air
is reactive, and the exhaust is nonreactive. Indeed, the fuel--air cycle computations (see
Figure 4.6) showed that efficiency increases with increasing residual fraction.
Emission Control Techniques for Lean Combustion Engines
With lean combustion engines, such as diesel and natural gas engines, catalytic converters
can be used to oxidize the HC and CO, but reduction of the exhaust nitric oxides is poor
because of the high oxygen content of the exhaust gases. Two nitric oxide control techniques
that have been developed for lean combustion conditions are selective catalytic reduction
(SCR) and the lean NO𝑥 trap.
Selective catalytic reaction uses liquid ammonia (NH3 ) or urea (CO(NH2 )2 ) sprayed
into the exhaust stream. As this mixture flows over a Pt/Rh catalyst, the NO and NO2 are
reduced as indicated by reactions, as shown in Equations 8.21 and 8.22:
4NO + 4NH3 + O2 ⟶ 4N2 + 6H2 O
(8.21)
2NO2 + 4NH3 + O2 ⟶ 3N2 + 6H2 O
(8.22)
This technique was developed by the Englehard Corporation in 1957. For vehicular
applications, a mixture of about 32% urea and 68% water, called DEF (diesel exhaust fluid)
is typically used, with an injection rate of about 2% of the diesel fuel flow rate. Use of
ammonia injection systems is limited to large marine engines and large-bore stationary
engines due to the toxicity of ammonia.
The lean NO𝑥 trap technique operates in a cyclic fashion, in which the trap adsorbs
and stores NO𝑥 as a nitrate during lean burn operation, and releases it as molecular nitrogen
during a short fuel-rich reducing desorption process. The NO𝑥 trap is composed of alkali or
alkaline earth materials, such as barium carbonate, that absorb and form nitrate species on
the surface of the catalyst during the adsorption process. Once these nitrate species saturate
the catalyst, the catalyst is regenerated by operating the engine in a rich condition to reduce
the NO2 to N2 . After regeneration, the alkali or alkaline earth materials are again available
for NO2 trapping.
258
Emissions
During the lean burn adsorption process, barium nitrate is formed from barium carbonate, as indicated by reaction, as shown in the Equation 8.23 below:
BaCO3 + 2NO2 ⟶ Ba(NO3 )2 + CO
(8.23)
During the short fuel-rich desorption step, excess fuel is used to produce molecular
nitrogen as indicated by reactions shown in the Equations 8.24 and 8.25:
Ba(NO3 )2 ⟶ BaO + 2NO2 + CO + O
(8.24)
2NO2 + 2CO ⟶ N2 + 2CO2
(8.25)
The engine control system controls the cycling rate between lean and periodic-rich
burn conditions to achieve a desired NO𝑥 conversion efficiency. Lean NO𝑥 traps have been
used on diesel, gasoline, and natural-gas-fueled engines, in which the engine fuel was used
as a fuel source for the rich reducing step, so this technique has a fuel penalty. Additional
information about the lean NO𝑥 trap technique is given by West et al. (2004), and diesel
emission reduction is given in the review by Johnson (2011).
8.7 REFERENCES
CHENG, W.K., D. HAMRIN, J. HEYWOOD, S. HOCHGREB, K. MIN, and M. NORRIS (1993), ‘‘An Overview
of Hydrocarbon Emissions Mechanisms in Spark Ignition Engines,’’ SAE paper 932708.
DANIEL, W. (1957), ‘‘Flame Quenching at the Walls of an Internal Combustion Engine,’’ Sixth
Symposium (International) on Combustion, p. 886, Reinhold, New York.
FENIMORE, C. (1971), ‘‘Formation of Nitric Oxide in Premixed Hydrocarbon Flames,’’ Thirteen
Symposium (International) on Combustion, the Combustion Institute (Pittsburgh), pp. 373--379.
GERRISH, H. and J. MEEM (1943), ‘‘The Measurement of Fuel Air Ratio by Analysis of the Oxidized
Exhaust Gas,’’ NACA report 757.
GILLETTE, A. and C. FERGUSON (1983), ‘‘Measurement and Analysis of the Particulate Emission from
a Direct Injection Diesel,’’ Particulate Sci. and Tech., Vol. 1, No. 1, pp. 77--90.
GREEVES, G., I. KHAN, C. WANG, and I. FENNE (1977), ‘‘Origins of Hydrocarbon Emissions from
Diesel Engines,’’ SAE paper 770259.
HANSON, R. and S. SALIMIAN (1984), ‘‘Survey of Rate Constants in the N/H/O System,’’ Chapter 6 in
Combustion Chemistry (W. Gardiner, Jr., ed.), Springer-Verlag, New York, pp. 361--421.
HEYWOOD, J. (1976), ‘‘Pollutant Formation and Control in Spark Ignition Engines,’’ Prog. Energy
Combust. Sci., Vol. 1, pp. 135--164.
HULS, T. and H. NICKOL (1967), ‘‘Influence of Engine Variables on Exhaust Oxides of Nitrogen
Concentrations from a Multicylinder Engine,’’ SAE paper 670482.
JOHNSON, T. (2011), ‘‘Diesel Emissions in Review,’’ SAE Paper 2011-01-0304.
KAISER, E., J. LORUSSO, G. LAVOIE, and A. ADAMCZYK (1982), ‘‘The Effect of Oil Layers on
the Hydrocarbon Emissons from Spark Ignited Engines,’’ Combustion Sci. Tech., Vol. 28,
pp. 69--73.
KOMIYAMA, K. and J. HEYWOOD (1973), ‘‘Predicting NO𝑥 Emissions and Effects of Exhaust Gas
Recirculation in Spark-Ignition Engines,’’ SAE paper 730475.
KUMMER, J. (1981), ‘‘Catalysts for Automobile Emission Control,’’ Prog. Energy Combust. Sci.,
Vol. 6, pp. 177--199.
KURTZ, E., D. FOSTER, and D. MATHER (2000), ‘‘Parameters that Affect the Impact of Auxiliary Gas
Injection in a DI Diesel Engine,’’ SAE Paper 2000-01-0233.
LAVOIE G., J. HEYWOOD, and J. KECK (1970), ‘‘Experimental and Theoretical Study of Nitric Oxide
Formation in Internal Combustion Engines,’’ Combust. Sci. Technol., Vol. 1, pp. 313--326.
Homework
259
MAYER, W., D. LECHMAN, and D. HILDENS (1980), ‘‘The Contribution of Engine Oil to Diesel Exhaust
Particulate Emissions,’’ SAE paper 800256.
MILLER, J., and BOWMAN, C. (1989), ‘‘Mechanism and Modeling of Nitrogen Chemistry in Combustion,’’ Prog. Energy Combust. Sci., Vol. 15, pp. 287--338.
MONDT, J. (2000), Cleaner Cars: The History and Technology of Emission Control Since the 1960’s,
SAE International, Warrendale, Pennsylvania.
PISCHINGER, R. and W. CARTELLIERI (1972), ‘‘Combustion System Parameters and Their Effect Upon
Diesel Engine Exhaust Emissions,’’ SAE paper 720756.
RAINE, R., C. STONE, and J. GOULD (1995), ‘‘Modeling of Nitric Oxide Formation in Spark Ignition
Engines with a Multizone Burned Gas,’’ Combust. Flame, Vol. 102, pp. 241--255.
TABACZYNSKI, R., J. HEYWOOD, and J. KECK (1972), ‘‘Time Resolved Measurements of Hydrocarbon
Mass Flowrate in the Exhaust of a Spark Ignition Engine,’’ SAE paper 72112.
WENTWORTH, J. (1971), ‘‘Effect of Combustion Chamber Surface Temperature on Exhaust Hydrocarbon Concentration,’’ SAE paper 710587.
WEST, B., S. HUFF, J. PARKS, S. LEWIS, J. CHOI, W. PARTRIDGE, and J. STOREY (2004), ‘‘Assessing
Reductant Chemistry During In-Cylinder Regeneration of Diesel Lean NO𝑥 Traps,’’ SAE Paper
2004-01-3023.
WOOD, K., J. CIUPEK, R. COOKS, and C. R. FERGUSON (1982), ‘‘Characterization of Diesel Particulates
by Mass Spectrometry Including MS-MS,’’ SAE paper 821217.
YU, R., V. WONG, and S. SHAHED (1980), ‘‘Sources of Hydrocarbon Emissions from Direct Injection
Diesel Engines,’’ SAE paper 800049.
ZELDOVICH, Y. (1946), ‘‘The Oxidation of Nitrogen in Combustion and Explosions,’’ Acta Physiochim.
U.R.S.S., Vol. 21, pp. 577--628.
8.8 HOMEWORK
8.1 Consider an equilibrium mixture of exhaust gases composed of O2 , N2 , and NO.
Using the equilibrium constant 𝐾p equation for the reaction 12 O2 + 12 N2 ⇌ NO
(see Chapter 3), plot the equilibrium mole fraction of NO as a function of temperature from 1000 to 4000 K for (a) 𝑃 = 1 atm, and (b) 𝑃 = 40 atm.
8.2 The rate of change of nitric oxide mass fraction for a fluid element because of chemical
reaction is given by Equation 8.10. The mass fraction can also change because of
NO convected in and out of the fluid element. Consider the control volume shown in
Figure 8.24. Write an expression for the rate of change of nitric oxide mass fraction
for this element assuming the fluid entering is devoid of nitric oxides, the fluid leaving
has the same properties as fluid in the element, and the generation of NO within the
control volume is given by Equation 8.10.
Control volume
.
mNO, generated
Figure 8.24 Illustration for Homework
Problem 8.2.
.
mNO, in
.
mNO, out
260
Emissions
8.3 The initial NO formation rate (mol/(cm3 s)) can be estimated (Heywood, 1976) given
the equilibrium (e) concentrations (mol/cm3 ) of oxygen and nitrogen, from the relation
𝑑[NO] 6 × 1016
exp
=
𝑑𝑡
𝑇 1∕2
(
−69, 090
𝑇
)
1∕2
(8.26)
[O2 ]𝑒 [N2 ]𝑒
Plot the exponential dependence of NO formation on temperature by computing the
formation rate over a range of temperatures from 1000 to 3000 K at 𝑃 = 10 bar and
equilibrium mole fractions of 𝑦N2 = 0.79 and 𝑦O2 = 0.20.
8.4 A CFR engine is operated with methane using the conditions given in Example 8.1.
The start of heat release is −15◦ atdc and the burn duration is 40◦ . The inlet conditions
are 𝑃1 = 100 kPa and 𝑇1 = 350 K. Using the Homogeneous.m program, plot the burn
fraction, pressure, burned and unburned temperatures, equilibrium and rate-limited
NO values as a function of crank angle.
8.5 A CFR engine is operated with gasoline using the conditions given in Example 8.1.
The start of heat release is −25◦ atdc and the burn duration is 60◦ . The inlet conditions
are 𝑃1 = 100 kPa and 𝑇1 = 350 K. Using the Homogeneous.m program, plot the
maximum burned temperature 𝑇b and the exhaust NO concentration as a function
of compression ratio. Vary the compression ratio from 7 to 12. Is engine knock a
concern under these conditions?
8.6 What are the equilibrium constants 𝐾p at 650 K for the two ammonia reactions given
by Equations 8.21 and 8.22?
8.7 A large-bore two-stroke stationary engine uses an ammonia injection system to remove NO from the exhaust. The engine has a displacement of 300 L and speed of
400 rpm. It is fueled with methane, operates lean at an equivalence ratio of 0.80, and
has a delivery ratio of 0.92. If 0.1% of the nitrogen entering the exhaust has been
converted to NO during the combustion process, what is the mass flow (kg/h) of
ammonia needed to reduce the NO to N2 ?
8.8 Use the equilibrium combustion solver program ecp.m to compute the exhaust CO
concentration for an engine fueled with C7 H17 . Plot the CO concentration versus 𝜙
for two different gas temperatures at the time of exhaust valve opening, 𝑇 = 1800 K
and 𝑇 = 1500 K. Assume an exhaust pressure of 200 kPa.
.
Ql
T, xHC
.
m out; x HC, out
C.V.
.
m in; x HC, in
Figure 8.25 Illustration for Homework
Problem 8.9.
Homework
261
8.9 Reaction of hydrocarbons in the exhaust port of an engine is an important process
since it alters the HC emission levels from gasoline and diesel engines. The rate of
change of the HC mass fraction due to chemical reaction is given by
)
(
𝑑𝑥
−𝐸
− HC = 𝐴 𝑥HC 𝑥O2 exp
𝑑𝑡
𝑅𝑇
Assuming that gases in the port shown in Figure 8.25 are well mixed, show that
)
(
)
(
𝑑𝑥HC
𝑚̇
−𝐸
= in (𝑥HC,in − 𝑥HC ) − 𝐴 𝑥HC 𝑥O2 exp
𝑑𝑡
𝑚
𝑅𝑇
c.v.
8.10 As an engine warms up, clearance between various parts change because of differing
amounts of thermal expansion. Explain how this can affect hydrocarbon emissions
from a spark ignition, homogeneous charge engine.
8.11 Explain how blowby can affect hydrocarbon exhaust emissions (not crankcase emissions that are no longer a problem). Specifically discuss the influence of engine
speed.
Chapter
9
Fuels
9.1 INTRODUCTION
So far our attention has been on fuels composed of only one chemical species. However,
a typical gasoline or diesel fuel may consist of 100 different hydrocarbons and another
100--200 trace species. In this chapter, we discuss why fuels are so complex, how they are
manufactured, their thermodynamic properties, and how they perform in engine applications.
Gasoline and diesel fuels for internal combustion engines are primarily obtained by
distillation from petroleum oil. Petroleum oil has a relatively low cost and a high-energy
density. It is a fossil fuel composed from ancient organic materials. Formation of petroleum
and natural gas reservoirs occurs underground during the pyrolysis of hydrocarbons in a
variety of endothermic reactions at high-temperature and/or pressure. Wells are drilled into
oil reservoirs to extract the crude oil. In 1858, Edwin Drake drilled the first U.S. oil well, a
21-m deep well in Titusville, Pennsylvania. He is credited with inventing the technique of
drilling inside a pipe casing to prevent water seepage. Innovations in the technology for oil
recovery have allowed deeper and deeper wells to be drilled. For example, oil is currently
pumped from reservoirs about 3000 m below the North Sea seabed in Europe.
The petroleum industry classifies crude oil by its geographical origin, its API (American Petroleum Institute) gravity (light or heavy), and its sulfur content (low sulfur is labeled
as sweet, and high sulfur is labeled as sour). Light crude oil produces a higher gasoline
fraction. Sweet crude oil is more valuable than sour crude oil because it requires less
refining to meet sulfur standards.
The identified worldwide crude oil reserves are estimated by the American Petroleum
Institute to be about 1 trillion barrels, with 0.6 trillion barrels remaining to be identified. At present consumption rates, at about 30 billion barrels per year, it is estimated
that petroleum reserves will last for 60--95 years. Technological advances in extraction
have created continual increases in the size of the worldwide petroleum reserves. For example, in 1950 the identified worldwide petroleum reserves were estimated to be about
0.09 trillion barrels, so in the last 60 years the identified petroleum reserves have increased
tenfold. To put the consumption of petroleum into perspective, about 0.7 trillion barrels of
petroleum have been consumed since the advent of the industrial revolution. The current
U.S. production of crude oil is about 10 million barrels per day. The recent invention and
commercialization of hydraulic fracturing, commonly known as ‘‘fracking’’, has enabled
greater production of petroleum and natural gas from underground shale formations.
Since petroleum contains carbon, its combustion produces carbon dioxide, a greenhouse gas linked to global warming. There are a number of private and governmental
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
262
Hydrocarbon Chemistry
263
initiatives underway to reduce the amount of greenhouse gas emissions from internal combustion engines. These initiatives include increased combustion and process efficiency, and
increased use of biofuels. The price of crude oil is dependent on geopolitical factors, and
has risen over the last 50 years to a maximum of about about $100 U.S. per barrel. At that
price level, alternative fuels such as biodiesel are becoming economically competitive.
The earliest internal combustion engines in the late 1800s were fueled with coal gas.
Coal gas is obtained by the coking, that is, partial pyrolysis of coal, similar to the process
of producing charcoal from wood. The pyrolysis process drives off the volatile constituents
in the coal. Coal gas is typically 50% hydrogen, 35% methane, 10% carbon monoxide,
and other trace gases such as ethylene. Coal gas was the primary source of gaseous fuel
in the United States until replaced by natural gas in the 1940s. Use of gaseous fuels such
as methane for internal combustion engines is increasing, due to increased availability and
relatively lower emissions relative to liquid fuels.
9.2 HYDROCARBON CHEMISTRY
Gasoline and diesel fuels are composed of blends of hydrocarbons, grouped into families of
hydrocarbon molecules termed paraffins, olefins, naphthenes, and aromatics. The hydrocarbon families each have characteristic carbon--hydrogen bond structures and chemical
formulae.
Paraffins (alkanes) are molecules in which carbon atoms are chained together by single
bonds. The remaining bonds are with hydrogen. They are called saturated hydrocarbons
because there are no double or triple bonds. The general formula for the paraffin family is
C𝑛 H(2𝑛+2) . The number of carbon atoms is specified by a prefix:
1-meth
5-pent
9-non
2-eth
6-hex
10-dec
3-prop
7-hept
11-undec
4-but
8-oct
12-dodec
Paraffin is designated as an alkane by the suffix -ane. Examples of paraffins are
methane, CH4 , and octane, C8 H18 , as shown schematically in Figure 9.1. Compounds with
straight chains are also labeled as normal or 𝑛-. For example, octane is sometimes called
normal octane or 𝑛-octane. Isooctane, shown in Figure 9.1, is an example of a branched
chain isomer of octane. That is, it has the same number of carbon atoms as octane but not
in a straight chain. The group CH3 attached to the second and fourth carbons from the right
is called a methyl radical, meth because it has one carbon atom and yl because it is of the
alkyl radical family C𝑛 H2𝑛+1 . Isooctane is more properly called 2,2,4-trimethylpentane,
2, 2, 4 because methyl groups are attached to the second and fourth carbon atoms, trimethyl
because three methyl radicals are attached, and pentane because the straight chain has five
carbon atoms.
Olefins (alkenes) are molecules with one or more carbon--carbon double bonds.
Monoolefins have one double bond, the general formula is C𝑛 H2𝑛 , and their names end
with -ene. For example, 1-octene, C8 H16 is shown in Figure 9.1. Isomers are possible not
only by branching the chain with the addition of a methyl radical but also by shifting the
position of the double bond without changing the carbon skeleton. Olefins with more than
one carbon--carbon double bond are undesirable components of fuel that lead to storage
problems. Consequently, they are refined out and the only olefins of significance in diesel
fuel or gasoline fuel are monoolefins.
264
Fuels
H
H
C
H
H
Methane, CH4
H
H
H
H
H
H
H
H
H
C
C
C
C
C
C
C
C
H
H
H
H
H
H
H
H
H
Octane, C8H18
CH3
CH3
CH3
CH2
C
CH
CH3
CH3
Isooctane, C8H18 or 2,2,4-trimethylpentane
(a)
H
H
H
H
H
H
H
C
C
C
C
C
H
H
H
H
H
C
C
C
H
H
H
H
1-Octene, C8H16
(b)
H
H
H
C
C
H
H
C
C
H
H
H
H H
C
H
C
H
Figure 9.1 (a) Paraffins, (b) olefins,
and (c) naphthenes.
H
C
H
Cyclopropane
Cyclobutane
(c)
Naphthenes (cycloalkanes) have the same general formula as olefins, C𝑛 H2𝑛 , but there
are no double bonds. They are called cyclo because the carbon atoms are in a ring structure.
Two examples are cyclopropane and cyclobutane, shown in Figure 9.1. Cycloalkane rings
having more than six carbon atoms are not as common.
Aromatics are hydrocarbons with carbon--carbon double bonds internal to a ring structure. The most common aromatic is benzene, shown schematically in Figure 9.2. Benzene
is a regulated toxic compound, as it is a known carcinogen. Notice that the double bonds
alternate in position between the carbon atoms. This makes the molecule’s bonds difficult to
break so that a greater temperature is required to initiate combustion. As a result, aromatics
are desirable in gasoline since they increase the octane number. Aromatics are undesirable
components of diesel fuels. Some common aromatics (toluene, ethylbenzene, and styrene)
have groups such as methyl radicals substituted for hydrogen atoms, and others (biphenyl)
have more than one ring. Finally, there are polycyclic aromatic hydrocarbons (PAH) that
are aromatics with two carbon atoms shared between more than one ring (naphthalene and
anthracene).
Hydrocarbon Chemistry
265
H
H
H
C
C
C
C
C
C
H
Equivalent
H
representation
H
Benzene
CH3
Toluene
CH2CH3
CH2
Ethylbenzene
CH2
Styrene
Biphenyl
Napthalene
Aromatics.
Figure 9.2
Anthracene
An alcohol is a partially oxidized hydrocarbon, formed by replacing a hydrogen atom
with the hydroxyl radical OH. If the hydrogen atom attached to an aromatic ring is replaced
by the hydroxyl radical, the molecule is called a phenol. Ethers are isomers of alcohol with
the same number of carbon atoms. Some examples, shown in Figure 9.3, are methanol,
ethanol, phenol, and methyl ether.
H
H
C
OH
H
H
Methanol
H
H
C
C
H
H
OH
OH
Ethanol
Phenol
(a)
H
C
H
H
H
H
O
C
H
H
C
H
H
Methyl ether
O
C
CH3
CH3
Methyl tertiary butyl ether (MTBE)
(b)
H
H
C
NO2
H
Figure 9.3 (a) Alcohols, (b) ethers,
and (c) nitroparaffins.
CH3
Nitromethane
(c)
266
Fuels
Nitromethane, CH3 NO2 , is formed from a paraffinic hydrocarbon by replacing a
hydrogen atom with a NO2 group, as shown in Figure 9.3. It has twice the bound oxygen as
monohydric alcohols, and can combust without air. At ambient temperature, it is a liquid,
and it is widely used as a drag racing fuel.
9.3 REFINING
Crude oil contains a large number of various hydrocarbon fractions. For example, 25,000
different compounds have been found in a sample of petroleum-derived crude oil. The
compounds range from gases to viscous liquids and waxes. The purpose of a refinery is to
physically separate crude oil into various fractions, and then chemically process the fractions
into fuels and other products. A full-scale crude oil refinery produces fuels for engines
(gasoline, diesel, and jet), fuels for heating (burner, coke, kerosene, and residual), chemical
feedstock (aromatics and propylene), and asphalt. On an average, a refinery will refine about
40% of the input crude oil into gasoline, 20% into diesel and heating fuel, 15% into residual
fuel oil, 5% into jet fuel, and the remainder into the other listed hydrocarbons. The fraction
separation process is called distillation and the device employed is a fractionating column.
The generic features of a small-scale fractionating column or still are illustrated in
Figure 9.4. The fractions at the top of the column have lower boiling points than the fractions
at the bottom. The column is heated preferentially, boiling off the lighter components. The
classification of the various fractions is arbitrary. In the order in which they leave the still,
the various fractions are commonly referred to as naphtha, distillate, gas oil, and residual
oil. Further subdivision uses the adjectives light, middle, or heavy. The adjectives virgin
or straight run are often used to signify that no chemical processing has been done to the
fraction. For example, since light, virgin naphtha can be used as gasoline, it is often called
straight run gasoline. The physical properties of any fraction depend on the distillation
temperatures of the products collected.
Water in
Condenser
Sample
Water out
Figure 9.4 Distillation process.
Fuel Properties
267
A broad cut fraction is collected over a large range of distillation temperatures, a narrow
cut over a small range, a light fraction over a low-temperature range, and a heavy fraction
over a high-temperature range. Gasoline fuel is a blend of hydrocarbon distillates with a
range of boiling points from about 25◦ --225◦ C, and diesel fuel is a blend of hydrocarbon
distillates with a range of boiling points from about 180◦ --360◦ C.
Chemical processing is required to convert one fraction into another. For example, a
crude might yield, on an energy basis, 25% straight run gasoline but the product demand
could be 50%. In this situation, the other 25% would be produced by chemical processing
of some other fraction into gasoline. Chemical processing is also used to upgrade a given
fraction. For example, straight run gasoline might have an octane number of 70, whereas
the product demand could be for an octane number of 90. In this case, chemical processing
would be needed to increase the octane number from 70 to 90.
Alkylation is used to increase the molecular weight and octane number of gasoline by
adding alkyl radicals to a gaseous hydrocarbon molecule. Light olefin gases are reacted
with isobutane using a catalyst. Isooctane results from reacting butene with isobutane. This
process requires relatively low-temperature (275 K) and pressure (300 kPa) and therefore
consumes relatively less energy than other refining processes.
Catalytic cracking uses activated catalysts to break the molecular chains of a distillate
to produce naphthas. Naphtha is an liquid mixture consisting of straight-chained and cyclic
aliphatic hydrocarbons having from five to nine carbon atoms per molecule. The naphtha
products of catalytic cracking are blended with other hydrocarbons to produce high octane
number gasolines. The reactions occur at high-temperature (700--800 K) and at low to
moderate pressure (200--800 kPa). Considerable energy is consumed in the process.
Reforming refers to reactions designed to alter molecular structure to yield higher
octane gasoline (e.g., conversion of paraffins into aromatic hydrocarbons). This is often
done using catalysts in a hydrogen atmosphere at high-temperature (800 K) and high
pressure (3000 kPa). Considerable hydrogen is produced as a result of the reaction:
C𝑛 H2𝑛+2 → C𝑛 H2𝑛−6 + 4H2
Coking is the process used to convert heavy reduced crude fraction to the more usable
naphtha and distillate fractions. The reduced crude is heated in an oven. Upon heating,
the molecules undergo pyrolytic decomposition and recombination. The average molecular
weight of the fraction remains the same, but a greater spectrum of components is produced.
The heaviest component, called coke, is a solid carbon material similar to charcoal.
9.4 FUEL PROPERTIES
The thermophysical properties, that is, enthalpy, specific heat, and entropy, of some single
hydrocarbons were given in Chapter 3. In general, the equivalent chemical formula Ca Hb
of a hydrocarbon--fuel mixture can be determined given the molecular weight 𝑀 and the
hydrogen to carbon ratio HC of the fuel:
𝑎 = 𝑀∕(12.01 + 1.008 HC) and
𝑏 = HC ⋅ 𝑎
Using the first law, the enthalpy of formation, ℎ𝑜f , at 298 K for a hydrocarbon fuel of
formula Ca Hb can be determined from the heat of combustion 𝑞c and the product enthalpies,
Equation 9.1:
[
]
𝑏
ℎf,H2 O − (1 − 𝜒)ℎfg,H2 O
(9.1)
ℎ𝑜f = 𝑞c + 𝑎ℎf,CO2 +
2
Fuels
5.0
Specific heat (kJ/(kg K))
4.2
3.4
2.5
Paraffins C5 to C20
1.6
Monoolefins C5 to C20
Aromatics C6 to C20
0.8
300
Figure 9.5 Specific heat of
various hydrocarbons.
500
700
900
1100
Temperature (K)
1300
1500
where 𝜒 is the quality of water in the products. The lower heat of combustion assumes
𝜒 = 1.0, whereas the higher heat of combustion assumes 𝜒 = 0.
Figures 9.5 and 9.6 show the ideal gas constant pressure specific heat, 𝑐𝑝,i of the
hydrocarbon constituents (paraffins, monoolefins, aromatics, naphthenes, and alcohols)
found in fuels as a function of temperature. The figures show that on a per unit mass basis
the specific heat depends on temperature and carbon type and is a weak function of carbon
number. This is not unexpected since the specific heat of a molecule depends on the number
and type of bonds. Assuming the specific heat per bond is constant, one expects the specific
heat to increase with the number of bonds, and on a per mole basis this is true. However,
since both the number of bonds and the mass of the molecule scale with the number of
carbon atoms, the specific heat is nearly constant on a per unit mass basis.
5.0
4.2
Specific heat (kJ/(kg K))
268
3.4
2.5
1.6
Naphthenes C5 to C20
0.8
Alkanols C1 to C22
Figure 9.6 Specific heat of
various hydrocarbons.
300
500
700
900
1100
Temperature (K)
1300
1500
Gasoline Fuels
269
Table 9.1 Specific Heat Curve-Fit Coefficients for Fuel Components
Type
𝑖
𝑎i
𝑏i
𝑐i
Paraffins
Monoolefins
Aromatics
Naphthenes
Alkanols
1
2
3
4
5
0.33
0.33
0.21
0.04
0.50
5.0
4.6
4.2
5.0
3.3
−1.5
−1.3
−1.3
−1.4
−0.71
The results shown are correlated by Equation 9.2:
𝑐𝑝,i = 𝑎i + 𝑏i 𝑡 + 𝑐i 𝑡2
(9.2)
where 𝑡 = 𝑇 (K) /1000, and 300 < 𝑇 < 1500 K. The 𝑎i , 𝑏i , 𝑐i coefficients of Equation 9.2
are listed in Table 9.1 for various hydrocarbon constituents. The specific heat, 𝑐p of a motor
fuel is
∑
𝑥i 𝑐𝑝,i
(9.3)
𝑐p =
i
where 𝑥i is the mass fraction of component 𝑖.
The absolute entropy (kJ/(kmol K)) of a liquid hydrocarbon fuel with formula
𝐶a 𝐻b 𝑂c 𝑁d has been correlated by Ikumi and Wen (1981):
𝑠 = 4.69𝑎 + 18.41𝑏 + 44.55𝑐 + 85.97𝑑
(9.4)
The octane numbers of various single hydrocarbon fuels are tabulated in Table 9.2.
In general, it has been found that the octane number is increased by reducing the straight
chain length. This can be accomplished by reducing the total number of carbon atoms or
by rearranging them into a branched chain structure. These generalizations are illustrated
in Figure 9.7. The critical compression ratio is determined by increasing the compression
ratio of a CFR engine (𝑁 = 600 rpm and inlet temperature 𝑇i = 311 K) until incipient knock
occurs. The correlation with octane number is evident. For further information about the
properties of internal combustion engine fuels, the reader is referred to Owen and Coley
(1995).
9.5 GASOLINE FUELS
Gasoline has been the dominant vehicular fuel since the early 1900s. It has a very high
volumetric energy density and a relatively low cost. It is composed of a blend of light
distillate hydrocarbons, including paraffins, olefins, naphthenes, and aromatics. It has a
hydrogen to carbon ratio varying from 1.6 to 2.4. A typical formula used to characterize
gasoline is C8 H15 , with a molecular weight of 111. A high hydrogen content gasoline is
C7 H17 .
Gasoline properties of interest for internal combustion engines are given in Table 9.3.
The properties include the octane number, volatility, gum content, viscosity, specific gravity, and sulfur content. The American Society for Testing and Materials (ASTM) has
established a set of gasoline specifications for each property, also listed in Table 9.3. The
antiknock index (AKI) is the average of the research (D2699) and motored (D2700) octane
numbers, and it is displayed on gasoline pumps at service stations (e.g., 85, 87, and 91).
270
Fuels
Table 9.2 Knock Characteristics of Single-Component Fuels
Formula
Name
CH4
C2 H6
C3 H8
C4 H10
C4 H10
C5 H12
C5 H12
C6 H14
C6 H14
C7 H16
C7 H16
C8 H18
C8 H18
C10 H12
C4 H8
C5 H10
C6 H12
C6 H12
C7 H14
C8 H16
C6 H6
C7 H8
C8 H10
C8 H10
C3 H6
C4 H8
C5 H10
C6 H12
C5 H8
C6 H10
C5 H8
CH4 O
C2 H6 O
Methane
Ethane
Propane
Butane
Isobutane
Pentane
Isopentane
Hexane
Isohexane
Heptane
Triptane
Octane
Isooctane
Isodecane
Methylcyclopropane
Cyclopentane
Cyclohexane
1,1,2-trimethylcyclopropane
Cycloheptane
Cyclooctane
Benzene
Toluene
Ethyl benzene
𝑚-Xylene
Propylene
Butene-l
Pentene-l
Hexene-l
Isoprene
1,5-Hexadiene
Cyclopentene
Methanol
Ethanol
Compression
ratio
Octane number
Research
12.6
12.4
12.2
5.5
8.0
4.0
5.7
3.3
9.0
3.0
14.4
2.9
7.3
120
115
112
94
102
62
93
25
104
0
112
−20
100
113
102
101
84
111
39
71
12.4
4.9
12.2
3.4
15
13.5
15.5
10.6
7.1
5.6
4.4
7.6
4.6
7.2
120
111
118
102
99
91
76
99
71
93
106
107
Motor
120
99
97
90
98
63
90
26
94
0
101
−17
100
92
81
95
78
88
41
58
115
109
98
115
85
80
77
63
81
38
70
92
89
Source: Obert, (1973).
For many years, the octane number of gasoline was above 90, and reached a maximum in
the 1960s, with leaded premium gasoline available with AKI ratings of 103+. As of 2014,
regular gasoline has a 87 AKI octane.
The octane number for aviation fuels is based on the motored (D2700) and supercharged
(D909) test methods.
Knowledge of gasoline volatility is important not only in designing fuel delivery
and metering systems, but also in controlling evaporative emissions. The volatility is
quantified by three related specifications: (1) the distillation curve (D86), (2) the Reid
vapor pressure (D323), and (3) the vapor--liquid ratio (D439). With the D86 distillation
Gasoline Fuels
271
14
Methane
Centralize
molecule
12
Add methyl
groups
11
Triptane
Critical compression ratio
10
Ethane
Lengthen
chain
9
Propane
8
7
Isooctane
6
n-Butane
5
4
n-Pentane
3
n-Hexane
n-Heptane
2
1
2
3
4
5
6
7
8
9
10
11
Number of carbon atoms in molecule
Figure 9.7 Effect of fuel structure on detonation tendency on paraffinic hydrocarbons (Lovell,
1948).
method, a still is used to evaporate the fuel. The fuel vapor is condensed at atmospheric
pressure. The heating rate is adjusted continuously such that the condensation rate is 4-5 mL/min. The heating process is stopped when the fuel starts to smoke and decompose,
typically around 370◦ C. The vapor temperature at the top of the distillation flask is measured
Table 9.3 Gasoline Property Specifications
Property
Benzene, vol%
Distillation, K
Gum, mg/mL
Heating value
Hydrocarbons, %
Octane, motored
Octane, research
Octane, supercharged
Reid vapor pressure, kPa
Specific gravity
Sulfur, wt%
ASTM method
D3606
D86
D381
D240
D1319
D2700
D2699
D909
D323
D287
D1266
272
Fuels
throughout the test. The volume fraction of condensate is plotted versus temperature to form
a distillation curve. The 10% and 90% evaporation temperatures, 𝑇10 and 𝑇90 , are used in
the volatility specifications. The 𝑇10 temperature, indicating the start of vaporization, is
used to characterize the cold starting behavior, and the 𝑇90 temperature, indicating the
finish of vaporization, is used to characterize the possibility of unburned hydrocarbons.
The ASTM drivability index (DI) is also a measure of fuel volatility and is defined in
Equation 9.5 as
DI = 1.5𝑇10 + 3𝑇50 + 𝑇90
(9.5)
Gum is a product of oxidation reactions with certain molecules often found in fuels. Use of
gasoline with a high gum component can lead to sticking of valves and piston rings, carbon
deposits, and clogging of fuel metering orifices. Inhibitors are often added to gasoline to
reduce the gum formed in such a test under an assumption that they will also reduce gum
formation in service. The ASTM D381 test method involves evaporating 50 mL of gasoline
in a glass dish at approximately 430 K by passing heated air over the sample for a period
of about 10 min. The difference in weight of the dish before and after the test is called the
existent gum content.
Reformulated Gasoline (RFG) and Renewable Fuel Standard (RFS)
The U.S. Clean Air Act of 1990 set up two programs, an oxygenated fuels program
and a reformulated gasoline (RFG) program, which resulted in mandated changes in the
composition of gasoline. The oxygenated fuels program is a winter program used to reduce
carbon monoxide and hydrocarbon levels in major cities that have carbon monoxide levels
that exceed federal standards. The oxygenated fuels program requires that gasoline contain
at least 2.7% by weight of oxygen. The first cities to use oxygenated gasoline were Denver,
CO, and Phoenix, AZ, and it is now required in about 40 cities in the United States.
The reformulated gasoline program is a year-round program used to reduce ozone
levels. It was mandated for metropolitan areas that have ozone levels that exceed federal
standards. The program requires that gasoline sold year-round in these areas have minimum
oxygen content of 2% by weight and maximum benzene content of 1%. It is now required
in 10 cities in the United States, and an additional 21 areas have voluntarily entered the
program. The primary oxygenate used is ethanol (EtOH). In 1996, California required use
of Tier 2 RFG, that has stricter standards than Tier 1 RFG.
The requirements for increased production volume of renewable fuels have greatly
expanded the market for biofuels. The U.S. Energy Policy Act of 2005 amended the Clean
Air Act and established a national renewable fuel standard (RFS) requiring that gasoline
contain 10% renewable fuels such as ethanol (E10). Studies are underway to allow use of
midrange blends such as E15 (15% ethanol) and E20 (20% ethanol) in standard gasolinefueled vehicles; however, there are fuel compatibility issues with the existing vehicle fleet.
The properties of various gasolines are compared in Table 9.4. The gasolines listed are
•
•
•
•
Industry average gasoline
Gasoline oxygenated with ethanol (gasohol)
Phase 1 reformulated gasoline
California Phase 2 reformulated gasoline
The volume percentage of olefins and benzene in reformulated gasoline is lower than
industry average gasoline. The Reid vapor pressure is reduced in the summer in reformulated
gasoline to reduce the emissions due to fuel evaporation. The 90% distillation temperature
Gasoline Fuels
273
Table 9.4 Properties of Gasoline Fuels
Average gasoline
Aromatics, vol%
Olefins, vol%
Benzene, vol%
Reid vapor pressure, kPa
(S: summer and W: winter)
𝑇50 , K
𝑇90 , K
Sulfur, mass ppm
Ethanol, vol%
Gasohol
Phase 1
RFG
23.9
8.7
1.6
67-S
79-W
367
431
305
10
23.4
8.2
1.3
50-S
79-W
367
431
302
4
28.6
10.8
1.60
60-S
79-W
370
440
338
0
Phase 2
RFG
25.4
4.1
0.93
46
367
418
31
0
Source: Adapted from EPA 420-F-95-007.
Table 9.5 FTP Regulated Emissions (g/mile) from
Industry Average and Reformulated Gasoline
Industry
average
gasoline
HC
NMHC
CO
NO𝑥
0.226
0.203
3.22
0.394
Phase 2
reformulated
gasoline
0.167
0.148
2.25
0.321
Source: Cadle et al., 1997.
T90 is decreased to increase the vaporization and oxidation of the gasoline, which reduces
the hydrocarbon emissions.
Since sulfur has an adverse impact on the performance of catalytic converters, the EPA
Tier 2 gasoline sulfur regulations, phased in from 2004 to 2007, reduced the sulfur level in
reformulated gasoline by 90% from about 300 to 30 ppm.
Table 9.5 compares the FTP regulated emissions from industry average gasoline and
Tier 2 reformulated gasoline for a group of fleet vehicles. The use of the reformulated
gasoline decreased the HC emissions by 26%, NMHC emissions by 27%, CO emissions
by 30%, and NO𝑥 emissions by 18%.
Gasoline Additives
Gasoline additives include octane improvers, anti-icers to prevent fuel line freeze-up, detergents to control deposits on fuel injectors and valves, corrosion inhibitors, and antioxidants
to minimize gum formation in stored gasoline. Alcohols, ethers, and methy cyclopentadienyl manganese tricarbonyl (MMT) are now used as octane improvers.
Many compounds have been tested for use as octane improvers in gasoline. Tetraethyl
lead was the primary octane improver in general use from 1923 to 1975. Its use in motor
vehicles was prohibited in 1995 due to its toxicity and adverse effect on catalytic converters
274
Fuels
and oxygen sensors. Currently, lead is only used in aviation and off-road racing gasolines.
Thomas Midgley (1889--1944), a mechanical engineer from the General Motors Research
Laboratory, discovered lead additives in 1921, as outlined in Midgley and Boyd (1922).
Midgley also was the inventor of Freon (F-12), a refrigerant initially developed for automotive air conditioning systems. Freon was the most widely used refrigerant in the world
until the mid-1990s when it was determined that the ultraviolet decomposition of Freon in
the stratosphere releases chlorine, causing depletion of the stratospheric ozone layer. The
manufacturing of Freon in the United States was prohibited in 1998.
9.6 ALTERNATIVE FUELS FOR SPARK IGNITION ENGINES
Important alternative fuels for spark ignition engines are compressed natural gas (CNG),
propane or liquid petroleum gas (LPG), alcohols, and hydrogen. Alternative fuels are of
interest, since they can be refined from renewable feedstocks, and their emission levels can
be much lower than those of gasoline and diesel-fueled engines (Dhaliwal et al., 2000).
If there are availability problems with crude oil, due to worldwide geopolitical problems,
alternative fuels can also be used as replacements. As of the year 2015, the most commonly
used alternative fuel for vehicles is propane, followed by natural gas, and methanol.
The cost of alternative fuels per unit of energy delivered can be greater than gasoline
or diesel fuel, and the energy density of alternative fuels by volume is less than gasoline
or diesel fuel. The smaller volumetric energy density requires larger fuel storage volumes
to have the same driving range as gasoline-fueled vehicles. This can be a drawback,
particularly with dual fuel vehicles, where a significant portion of the trunk space is used
by the alternative fuel storage tank. Alternative fuels also lack a wide-scale distribution
and fueling infrastructure comparable to that of conventional fuels. In recent years, fleet
vehicles, such as buses, trucks, and vans have been a growing market for alternative fuels,
as they can operate satisfactorily with localized fueling. In 1990, there were about 4 million
propane-fueled vehicles, 3 million ethanol-fueled vehicles, and about 1 million natural gasfueled vehicles worldwide, compared with about 150 million gasoline-fueled vehicles in
the United States alone (Webb and Delmas, 1991).
Existing gasoline or diesel engines can be retrofitted fairly easily for operation with
alternative fuels. However, various operational considerations need to be taken into account.
The different combustion characteristics of alternative fuels require a change in the injection
and ignition timing. Also, many alternative fuels, especially those in gaseous form, have
very low lubricity, causing increased wear of fuel components such as fuel injectors and
valves.
The properties of various alternative fuels are tabulated in Table 9.6, and are compared
with the properties of 𝑛-octane. The first three columns contain gaseous fuels (methane,
propane, and hydrogen) and the next three columns are liquid fuels (methanol, ethanol, and
𝑛-octane). While there is a range of energy densities on a fuel mass (MJ/kgfuel ) basis, the
energy densities are comparable on a stoichiometric air mass (MJ/kgair ) basis. Octane has
the greatest energy density by volume (MJ/L). Alternate fuels have higher octane levels
than gasoline, so engines fueled with alternative fuels can operate at higher compression
levels, and thus at higher efficiency. Further information about alternative fuels and their
use is given in Owen and Coley (1995).
The methane number is a measure of the tendency for a gaseous fuel to knock. As
indicated in Table 9.7, Malenshek and Olsen (2009) found a linear relationship between
the maximum compression ratio and the methane number, for a variety of gaseous fuels,
including coal gas, wood gas, digester gas, and landfill gas. A fuel’s methane number limits
Alternative Fuels for Spark Ignition Engines
275
Table 9.6 Thermodynamic Properties of Spark Ignition Fuels
Propane Natural gas Hydrogen Methanol Ethanol
Molecular weight
Boiling point (◦ C),
at 1 bar
Mass A/F ratio,
stoichiometric
Vapor pressure (kPa),
at 32◦ C
Enthalpy of vaporization,
ℎfg (kJ/kg), at 298 K
Adiabatic flame
temperature (K)
Vapor flammability
limits (% volume)
Lower heating value,
mass, (MJ/kgfuel )
Lower heating value,
volume, (MJ/kgfuel )
Lower heating value,
stoichiometric (MJ/kgair )
Octane number, research
Octane number, motor
Stoichiometric CO2
emissions, (g CO2 /MJfuel )
Gasoline
44.10
−42
18.7
−160
2.015
−253
32.04
65
46.07
78
∼110
30--225
15.58
17.12
34.13
6.43
8.94
15.04
32
17
62--90
1215
850
310
2268
2227
2383
2151
2197
2266
2.1--9.5
5.3--15
5--75
5.5--26
3.5--26
0.6--8
46.4
50.0
120
19.9
26.8
44.5
25.5
8.1
15.7
21.1
32.9
2.98
2.92
3.52
3.09
3.00
2.96
100
95.4
64.5
120
120
54.9
106
112
91
69
111
92
71.2
90--98
80--90
71.9
0
Source: Adapted from Black, 1991 and Unich et al., 1993.
Table 9.7 Critical Compression Ratio Versus Methane Number
Gas
Coal gas
Steam-reformed natural gas
Wood gas
Natural gas
Methane
Digester gas
Landfill gas
Compression ratio
Methane number
8.0
10.5
10.3
23.9
62.4
70.2
78--98
100
139.1
139.6
14.4
17.6
17.6
Source: Malenshek and Olsen, 2009.
the maximum compression ratio and thus the theoretical engine efficiency. For example, an
engine optimized to operate on natural gas with a methane number of about 90 is susceptible
to knock when operated on gases that have a lower methane number, such as coal gas which
has a methane number of 24. The octane number of methane is 120 (RON), one of the
highest values for hydrocarbon fuels.
276
Fuels
Propane
Propane (C3 H8 ), is a saturated paraffinic hydrocarbon. When blended with butane (C4 H10 )
or ethane (C2 H6 ), it is also designated as liquefied petroleum gas. A common LPG blend
is P92, which is 92% propane and 8% butane. In the United States, about one-half of the
LPG supply is obtained from the lighter hydrocarbon fractions produced during crude oil
refining, and the other half from heavier components of wellhead natural gas.
Propane has been used as a vehicular fuel since the 1930s. In 1993, there were about
4 million LPG vehicles operating worldwide, with the majority in the Netherlands, followed
by Italy, the United States, and Canada. There is a relatively extensive refueling network for
propane, with over 15,000 refueling stations available in North America. There are a number
of original equipment manufacturers that currently sell propane-fueled vehicles, primarily
light- and medium-duty fleet vehicles, such as pick-up trucks and vans. Conversion kits
are also available to convert gasoline or diesel-fueled engines to dedicated propane or dual
fuel use.
In vehicles, propane is stored as a compressed liquid, typically from 0.9 to 1.4 MPa. Its
evaporative emissions are essentially zero, since it is used in a sealed system. A pressure
regulator controls the supply of propane to the engine and converts the liquid propane to a
gas through a throttling process. Propane gas can be injected into the intake manifold, into
the ports, or directly into the cylinder. Propane has an octane number of 112 (RON), so
vehicular applications of propane can operate at a raised compression ratio.
As shown in Table 9.6, the CO2 emissions on an equivalent energy basis are about
90% that of gasoline. Liquid propane has three-fourths of the energy density by volume of
gasoline so that the fuel economy is correspondingly reduced. The volumetric efficiency
and the power are also reduced due to the displacement of about 5--10% of the intake air
by propane and the loss of evaporative charge cooling. Propane requires about a 5◦ spark
advance at lower engine speeds due to its relatively low flame speed.
Representative FTP emissions from an LPG-fueled engine are shown in Table 9.8.
The engine used was a 3.1 L engine with an LPG conversion system using an intake
manifold mixer. The LPG fuel used was HD5 propane (96% propane and 4% ethane). The
results indicate that the HC and CO emissions were lower with LPG than gasoline, 43 and
53% respectively, but the NO𝑥 levels were higher. The toxic emissions are also given in
Table 9.8 LPG-Fueled Vehicle (3.1 L Engine) Emissions
Propane
Gasoline
Regulated emissions (g/mile)
HC
CO
NO𝑥
0.21
2.55
0.67
0.37
5.4
0.42
Toxic emissions (mg/mile)
Benzene
1,3-Butadiene
Formaldehyde
Acetaldehyde
Total
<0.1
<0.1
1.2
0.3
1.5
16.7
2.5
3.1
1.5
23.8
Source: Bass et al., 1993.
Alternative Fuels for Spark Ignition Engines
277
Table 9.8. The levels of toxic emissions are typically an order of magnitude less than the
baseline gasoline toxic emissions.
Natural Gas
Natural gas is a naturally occurring fuel found in oil fields. It is primarily composed
of about 90--95% methane (CH4 ), with small amounts of additional compounds such as
0--4% nitrogen, 4% ethane, and 1--2% propane. Methane is a greenhouse gas, with a global
warming potential approximately 10 times that of carbon dioxide. As shown in Table 9.6,
since methane has a lower carbon to hydrogen ratio relative to gasoline, its CO2 emissions
are about 22--25% lower than gasoline.
Natural gas has been used for many years in stationary engines for gas compression
and electric power generation. An extensive distribution network of natural gas pipelines
exists to meet the need for natural gas for industrial processes and heating applications.
Natural gas-fueled vehicles have been in use since the 1950s, and conversion kits are
available for both spark and compression ignition natural gas and gasoline or diesel fuel.
As of 2013, there are about 18 million natural gas-fueled road vehicles worldwide, and the
number of natural gas-fueled vehicles is expected to double by 2020. One advantage of
a bifuel operation is that the operating range of a vehicle is extended in comparison with
a dedicated natural gas-vehicle. Currently, original equipment manufacturers are selling
production natural gas-fueled vehicles, primarily to fleet owners. Natural gas vehicles were
the first vehicles to meet the California ULEV emission standards.
Natural gas is stored in a compressed (CNG) state at room temperatures and also in a
liquid (LNG) form at 160◦ C. Natural gas has an octane number (RON) of about 120 so that
natural gas engines can operate at a compression ratio higher than that of gasoline-fueled
engines. Natural gas is pressurized to 20 MPa in vehicular storage tanks so that it has
about one-third of the volumetric energy density of gasoline. The storage pressure is about
20 times that of propane. Like propane, natural gas is delivered to the engine through a
pressure regulator, either through a mixing valve located in the intake manifold, port fuel
injection at about 750 kPa, or direct injection into the cylinder. With intake manifold mixing
or port fuel injection, the engine’s volumetric efficiency and power is reduced due to the
displacement of about 10% of the intake air by the natural gas, and the loss of evaporative
charge cooling. Natural gas does not require mixture enrichment for cold starting, reducing
potential cold start HC and CO emissions.
The combustion of methane is different from that of liquid hydrocarbon combustion,
since only carbon--hydrogen bonds are involved, and no carbon--carbon bonds, so the
combustion process is more likely to be more complete, producing less nonmethane hydrocarbons. Optimal thermal efficiency occurs at rich conditions with equivalence ratios
of 1.3--1.5. The total hydrocarbon emission levels can be higher than gasoline engines due
to unburned methane. The combustion of methane can produce formaldehyde, a regulated
toxic pollutant. The particulate emissions of natural gas are very low relative to diesel fuel.
Natural gas has a lower adiabatic flame temperature (2240 K) than gasoline (2310 K), due
to its higher product water content. Operation under lean conditions will also lower the
peak combustion temperature. The lower combustion temperatures lower the NO formation
rate, and produce less engine-out NO𝑥 .
To meet vehicular emission standards, catalytic converters are used with natural gasfueled engines. Since three way catalytic converters are most effective at stoichiometric
conditions, natural gas combustion needs to be maintained at stoichiometric, and exhaust
gas recirculation (EGR) is used to reduce the peak combustion temperatures and thus the
278
Fuels
Table 9.9 CNG-Fueled Vehicles (2.2 L Engine) Regulated Emissions (g/mile)
Emission
NMOG
CO
NO𝑥
Toyota engine
GMC engine
GMC engine
CNG
0.007
0.69
0.015
CNG
0.027
1.01
0.10
Gasoline
0.08
1.54
0.17
Source: Sun et al., 1998, Kato et al., 1999.
nitrogen oxide levels. Table 9.9 gives the exhaust emissions for a 2.2 L bifuel gasoline
and CNG engine, and a 2.2 L dedicated CNG engine. When the bifuel engine is switched
from gasoline to CNG, the nonmethane organic gases (NMOG), carbon monoxide (CO),
and nitrogen oxide (NO𝑥 ) levels were reduced to 60%, 34%, and 41%, respectively. The
dedicated CNG engine was modified to operate specifically with natural gas, with a higher
compression ratio, intake valves with early closing timing, and intake and exhaust valves
with increased lift.
The emissions of FTP toxics from a 0.75 ton light-duty truck operated with gasoline
and with natural gas are given in Table 9.10. The engine emission control system included
a heated oxygen sensor and a standard three-way catalyst. The same compression ratio of
8.3:1 was used for both fuels. Table 9.9--9.11 also indicate that the CNG toxic emissions are
much less than the gasoline toxic emissions. The highest mass emissions with gasoline were
benzene and formaldehyde, and the highest mass emissions with CNG was formaldehyde,
at a level about half of that of gasoline.
Natural gas can replace diesel fuel in heavy-duty engines with the addition of a spark
ignition system. A number of heavy-duty diesel engine manufacturers are also producing
dedicated natural gas heavy-duty engines. The natural gas-fueled engines are operated lean
with an equivalence ratio as low as 𝜙 = 0.7. The resulting lower in-cylinder temperatures
reduce the NO𝑥 levels. Heavy-duty natural gas engines are designed to meet LEV emission
standards without the use of an exhaust catalyst and will meet ULEV emission standards
with the addition of a catalyst. The emission certification data for three heavy-duty natural
gas engines are given in Table 9.11.
Natural gas can also be used in compression ignition engines if diesel fuel is used as a
pilot fuel, since the autoignition temperature of methane is 540◦ C, compared with 260◦ C
for diesel fuel. This fueling strategy is attractive for heavy-duty diesel applications, such
as trucks, buses, locomotives, and ships, compressors, and generators. These engines are
Table 9.10 CNG-Fueled Vehicles Toxic Emissions (mg/mile)
Toxic
CNG
CNG start/
gasoline run
Gasoline
Benzene
1,3-Butadiene
Formaldehyde
Acetaldehyde
Total
0.2
6 0.1
3.4
0.2
3.8
14.8
0.1
4.1
0.3
19.3
31.2
1.5
5.9
2.0
40.6
Source: Springer et al., 1994.
Alternative Fuels for Spark Ignition Engines
279
Table 9.11 Heavy-Duty Natural Gas Engine Emission Certification Data (g/bhp-h)
Hercules
GTA 5.6
Power (hp)
NMHC
CO
NO𝑥
PM
190
0.9
2.8
2.0
0.10
Cummins
L10
240
0.2
0.2
1.4
0.02
Detroit
Diesel 50G
275
0.9
2.8
2.6
0.06
Source: Owen and Coley, 1995.
also operated with a lean combustion mixture so that the NO𝑥 emissions are decreased.
However, since diesel engines are unthrottled, at low loads the lean combustion conditions
can degrade the combustion process, increasing the hydrocarbon and carbon monoxide
emissions.
Ethanol
Ethanol (C2 H5 OH) is an alcohol fuel formed from the fermentation of sugar and grain
stocks, primarily sugar cane and corn, which are renewable energy sources. Its properties
and combustion characteristics are very similar to those of methanol. Ethanol is also called
‘‘grain’’ alcohol. It is a liquid at ambient conditions and nontoxic at low concentrations.
Gasohol (E10) is a gasoline--ethanol blend with about 10% ethanol by volume. E85
is a blend of 85% ethanol and 15% gasoline. In Brazil, about half of the vehicles use
an ethanol-based fuel ‘‘alcool,’’ primarily E93, produced from sugar cane. In the United
States, the primary source of ethanol is currently from starch feedstocks, such as corn,
and there are efforts underway to produce ethanol from cellulosic feedstocks such as corn
fiber, forestry waste, poplar, and switch grass. The energy density by volume of ethanol is
relatively high for an alternative fuel, about two-thirds that of gasoline. The octane rating
of ethanol of 111 RON allows use of an increased compression ratio. The cetane number
of ethanol is low, at about 8, and it can be used in compression ignition engines with diesel
fuel pilot ignition.
As shown in Table 9.6, the CO2 emissions from ethanol on an equivalent energy basis
are about 99% that of gasoline. With a switch from RFG to E85, for a fleet of flexiblefueled vehicles, Cadel et al. (1997) report that the NO𝑥 emissions decreased by 29%, the
nonmethane hydrocarbons (NMHC) decreased by 10%, and the CO emissions increased
by 8%. The corresponding FTP toxic emissions are shown in Table 9.12. There was a
71% reduction in 1,3-butadiene, and a 64% reduction in benzene. However, for E85 the
acetaldehyde emissions were almost two orders of magnitude higher than those of RFG,
leading to almost a fourfold increase in the toxic emission levels.
Methanol
Methanol (CH3 OH) is an alcohol fuel formed from natural gas, coal, or biomass feedstock.
Methanol has been used as a vehicular fuel since the early 1900s, and is also used as a fuel
for diesel engines and fuel cells. It is also called wood alcohol. It is a liquid at ambient
conditions. Its chemical structure is a hydrocarbon molecule with a single hydroxyl (OH)
280
Fuels
Table 9.12 FTP Toxic Emissions (mg/mile) from Ethanol-Fueled Vehicles
Toxic
E85
Phase 2 RFG
Benzene
1,3-Butadiene
Formaldehyde
Acetaldehyde
Total
1.8
0.2
4.1
24.8
20.7
5.1
0.7
2.1
0.5
8.4
Source: Cadle et al., 1997.
radical. The hydroxyl radical increases the polarity of the hydrocarbon so that methanol
is miscible in water, and has a relatively low vapor pressure. Since oxygen is part of the
chemical structure, less air is required for complete combustion. Methanol is very toxic,
and ingestion can cause blindness and death.
Pure methanol is labeled M100, and a mix of 85% methanol and 15% gasoline is
labeled M85. M85 has an octane rating of 102. Adding gasoline to methanol provides more
volatile components that can vaporize more easily at low-temperatures. Methanol has been
adopted as a racing fuel, both for performance and safety reasons. Since methanol mixes
with water, a methanol fire can be extinguished with water, which is not the case with
gasoline. The octane rating of methanol M100 of 111 RON allows use of an increased
compression ratio. The relatively high enthalpy of evaporation (1215 kJ/kg) of methanol
relative to gasoline (310 kJ/kg) produces greater intake air-cooling and a corresponding
increase in volumetric efficiency relative to gasoline. The energy density by volume of
methanol is about half that of gasoline. However, because of its oxygen content, it has a
higher stoichiometric energy density (3.09 MJ/kg air) relative to gasoline (2.96 MJ/kg air).
For maximum power, a rich equivalence ratio of 𝜙 = 1.6 is used.
Flexible-fuel vehicles (FFV) have been developed to use a range of methanol and gasoline blends ranging from 100% gasoline to M85. An optical fuel sensor is used to determine
the alcohol content and adjust the fuel injection and spark timing. The engine compression ratio is not increased, to allow for the lower octane level of gasoline. The low vapor
pressure of methanol causes cold starting problems. Satisfactory cold starting with M85
requires a rich mixture so that enough volatiles are present to form a combustible mixture.
Methanol is corrosive, especially to rubber and plastic, so alcohol tolerant components,
such as stainless steel, are required for its storage and transport.
The cetane number of methanol is low at about 5, and it can be used in compression
ignition engines with diesel fuel pilot ignition. Methanol burns with a nearly invisible flame,
and a relatively high flame speed. Formaldehyde is a significant decomposition product
from methanol combustion and is expected to be higher from methanol than other fuels.
The formaldehyde emissions are proportional to the equivalence ratio, so rich combustion
will produce increased emissions of formaldehyde. Special lubricants also need to be used
in methanol-fueled engines.
As shown in Table 9.6, the CO2 emissions of methanol on an equivalent energy basis are
about 96% that of gasoline. With a change in the fuel for a fleet of flexible-fueled vehicles
from RFG to M85, the nonmethane hydrocarbons (NMHC) and CO emissions decreased
by 30% and 17% respectively, and the NO𝑥 emissions remained about the same (Cadle
et al., 1997). The FTP toxic emissions for the methanol- and gasoline-fueled-flexiblefueled vehicles are given in Table 9.13. There was an 83% reduction in 1,3-butadiene, a
50% reduction in benzene, and a 25% increase in acetaldehyde. However, for M85 the
Hydrogen
281
Table 9.13 Toxic Emissions (mg/mile) from Methanol-Fueled
Vehicles
Toxic
M85
Phase 2 RFG
Benzene
1,3-Butadiene
Formaldehyde
Acetaldehyde
Total
3.0
0.10
17.1
0.5
20.7
6.0
0.6
1.6
0.4
8.6
Source: Cadle et al., 1997.
formaldehyde emissions were almost an order of magnitude higher than those of RFG,
leading to more than a twofold increase in the toxic emission levels.
9.7 HYDROGEN
Hydrogen (H2 ) can be produced from many different feedstocks, including natural gas,
coal, biomass, and water. The production processes include steam reforming of natural gas,
presently the most economical method, electrolysis of water, and gasification of coal, which
also produces CO2 . Hydrogen is colorless, odorless, and nontoxic, and hydrogen flames
are invisible and smokeless. The global warming potential of hydrogen is insignificant
in comparison to hydrocarbon-based fuels, since combustion of hydrogen produces no
carbon-based compounds such as HC, CO, and CO2 .
At present, the largest user of hydrogen fuel is the aerospace community for rocket fuel.
Hydrogen can also be used as a fuel in fuel cells. There have been a number of vehicular
demonstration projects, but the relatively high cost of hydrogen fuel has hindered adoption
as an alternative fuel. Dual fuel engines have been used with hydrogen, in which hydrogen
is used at start up and low load, and gasoline at full load (Fulton et al., 1993) to reduce the
cold start emissions levels.
One of the major obstacles related to the use of hydrogen fuel is the lack of any
manufacturing, distribution, and storage infrastructure. The most economical method would
be to distribute hydrogen through pipelines, similar to natural gas distribution. The three
methods used to store hydrogen are: (1) in a liquid form at −253◦ C in cryogenic containers,
(2) as a metal hydride, such as iron--titanium hydride FeTiH2 , or (3) in a pressurized gaseous
form at 20--70 MPa. The metal hydride releases hydrogen when heated by a heat source,
such as a vehicle exhaust system. The most common storage methods are liquid and hydride
storage, which have comparable volumetric storage capabilities, both requiring about 10
times the space required by an equivalent 5-gallon gasoline tank, as shown by Table 9.14.
At least a 55-gallon tank of compressed hydrogen is needed to store the energy equivalent
of 5 gallons of gasoline.
Compressed hydrogen at 70 MPa has one-third the energy density by volume of
compressed natural gas, and liquid hydrogen has one-fourth the energy density by volume of
gasoline. Use of liquid hydrogen has an additional energy cost, as liquefaction of hydrogen
to −20 K requires an expenditure of energy approximately equal to the energy content of
liquid hydrogen. If mixed with air in the intake manifold, the volume of hydrogen is about
30% of the intake mixture volume at stoichiometric, decreasing the volumetric efficiency.
The octane rating of hydrogen of 106 (RON) allows use of an increased compression ratio.
282
Fuels
Table 9.14 Comparison of Hydrogen Storage Methods
Energy (kJ)
Fuel mass (kg)
Tank mass (kg)
Total fuel system mass (kg)
Volume (gal)
Gasoline
(5 gallons)
Liquid H2
Hydride Fe
Ti (1.2%)
Compressed
(70 MPa) H2
6.64 × 105
14
6.5
20.5
5
6.64 × 105
5
19
24
47
6.64 × 105
5
550
555
50
6.64 × 105
5
85
90
60
Source: Kukkonen and Shelef, 1994.
The combustion characteristics of hydrogen are very different from gasoline combustion characteristics, as the laminar flame speed of a hydrogen--air mixture is about 3 m/s,
about 10 times that of methane and gasoline, and the adiabatic flame temperature is about
100◦ C higher than gasoline and methane. Since it has a wide flammability limit (5--75%),
preignition and backfiring can be a problem. The flammability limits correspond to equivalence ratios of from 0.07 to 9.0. Water injection into the intake manifold is used to mitigate
preignition and provide cooling. Exhaust gas recirculation and lean operation are used to
reduce NO𝑥 levels.
9.8 DIESEL FUELS
Diesel fuel consists of a mixture of light distillate hydrocarbons that have molecular weights
from about 170 to 200, with a corresponding range of 12--20 carbon atoms per molecule.
Diesel fuels vaporize in the range between about 180 and 360◦ C, higher than gasoline. It
is estimated that there are more than 10,000 isomers in diesel fuel. Like gasoline, diesel
fuels are mixtures of paraffinic, olefinic, naphthenic, and aromatic hydrocarbons, but their
relative proportions are different. Diesel fuels have about an 8% greater energy density by
volume than gasoline, and are the primary fuel used by heavy-duty vehicles. Diesel fuel
is also much less flammable than gasoline, as the flash point temperature of diesel 2-D is
52◦ C, much higher than that of gasoline, which is about −40◦ C. The flash point is defined
as the lowest temperature at which a vapor--air mixture will ignite when an ignition source
is applied.
Diesel fuels are classified both by a numerical scale and by use. The use designations
are bus, truck, railroad, marine, and stationary. The American Society for Testing and
Materials, ASTM D975, numerical classification scheme for diesel fuels ranges from one
to six, with letter subcategories. Diesel fuel number 1-D is a light (approximated by
C12 H22 ) distillate cold weather fuel with a flash point of 38◦ C. Diesel fuel 2-D is a middle
(approximated by C15 H25 ) distillate diesel fuel of lower volatility and is the most common
fuel for vehicular applications. Diesel fuel 4-D is a heavy distillate fuel used for stationary
applications where the engine speed is low and more or less constant. It has a flash point
of 55◦ C. As the number designation increases, the mean molecular weight and viscosity of
the fuel increases.
The specification chart contained in ASTM D975 is shown here as Table 9.15. These
specifications are used by refiners as a basis for the control of diesel fuel compositions.
The corresponding European Standards Organization (CEN) standard for diesel fuel is EN
590, listed in Table 9.16. The thermodynamic properties of diesel fuel 2-D are listed in
Diesel Fuels
283
Table 9.15 Diesel Fuel Specifications (ASTM D975)
ASTM
Method
Minimum cetane number
Minimum flash point, ◦ C
Cloud point, ◦ C
Maximum water and sediment, vol%
Maximum carbon residue
Maximum ash, wt%
𝑇90 , K
Kinematic viscosity at 40 ◦ C (mm2 /s)
Maximum copper strip corrosion
D613
D93
D2500
D524
D482
D86
D445
No. 1-D
No. 2-D
No. 4-D
40
38
Local
0.05
0.15
0.01
561 max
1.3--2.4
No. 3
40
52
Local
0.05
0.35
0.01
555--611
1.9--4.1
No. 3
30
55
Local
0.05
0.10
5.5--24
Table 9.18. In CFD simulations, diesel fuel is often approximated by tetradecane (𝑛-C14 H30 )
for computation of evaporation, ignition, and combustion.
One component of diesel fuel that has attracted particular regulatory attention is sulfur,
due to the adverse impact that sulfur-containing particulate emissions have on air quality.
), and sulfur dioxide
During combustion the sulfur in the diesel fuel forms sulphates (SO−
4
(SO2 ), which reacts with water to form sulfuric acid, a component of acid rain. Ultralow
sulfur diesel (ULSD) has a sulfur content of 15 ppm or less. As part of the Euro V emission
standards, ULSD is required in Europe. After December 1, 2014, all highway, nonroad,
locomotive, and marine diesel fuel produced and imported in the United States will be
mandated to be ultralow sulfur diesel. Sulphur reacts with the nickel in metal alloys in fuel
injectors to form a eutectic alloy with increased lubricity. Therefore, with decreased sulfur
concentration, additional fuel additives will be needed to maintain the lubricity of diesel
fuel.
At low-temperatures, the higher molecular weight components in diesel fuel have a
tendency to jell or crystallize into solid wax particles giving it a cloudy appearance. The
onset of this crystallization is called the cloud point. A standardized test is used to specify
the cold filter plugging point (CFPP) temperature for various grades of diesel fuel. Additives
are used to modify the crystal structure of the wax particles to reduce the plugging of the
fuel filter at low-temperatures.
The cetane numbers of various hydrocarbons used in diesel fuel are listed in Table
9.17. The octane number and the cetane number of a fuel are inversely correlated. As shown
in Figure 7.18, when the octane number is decreased from 85 to 0, the corresponding cetane
number increases from 20 to 60. Therefore, diesel fuel is a poor spark ignition fuel and
vice versa. Since a low cetane number fuel will mix more completely with the cylinder air
before burning, the local equivalence ratio of the initial premixed burn will be less (𝜙 ∼ 3)
than the local equivalence ratio (𝜙 ∼4) for a higher cetane number.
There are a number of correlations used to approximate the ignition quality of a diesel
fuel. One widely used approximation is the calculated cetane index, CCI. The CCI is
an approximation to the cetane number using the ASTM D976 empirical correlation for
petroleum-based diesel fuels, and is based on the API gravity and the midpoint (50%
evaporated) boiling temperature.
2
CCI = −420.34 + 0.016 𝐺2 + 0.192 𝐺 log 𝑇50 + 65.01 (log 𝑇50 )2 − 0.0001809 𝑇50
284
Fuels
Table 9.16 European Diesel Fuel Specifications (EN 590)
Temperate
climate
Minimum cetane number
Minimum cetane index
Density at 15◦ C, (kg/m3 )
Polycyclic aromatic hydrocarbons, %
Minimum flash point, (◦ C)
Maximum water, (mg/kg)
Total contamination, (mg/kg)
Maximum ash, wt%
Distillation recovered 95%(v/v), (◦ C)
Kinematic viscosity at 40◦ C (mm2 /s)
Maximum sulfur, ppm
Maximum copper strip corrosion
Lubricity, wear scar diameter (wsd), (𝜇m )
51
46
820--845
11
55
200
24
0.01
360
2.00--4.50
10
Class 1
460
ISO
Test method
51-65
42-64
3675
12916
2719
12937
12662
6245
3405
3104
20884
2160
12156
Table 9.17 Centane Numbers of Hydrocarbon Fuels
Formula
Name
C16 H32
C7 H16
C8 H18
C12 H26
C14 H30
C16 H34
Heptamethylnonane
𝑛-Heptane
𝑛-Octane
𝑛-Dodecane
𝑛-Tetradecane
𝑛-Hexadecane (cetane)
Cetane number
15
56
64
88
96
100
Source: Hurn and Smith (1951).
Table 9.18 Comparison of Thermodynamic Properties of Compression Ignition Fuels
Formula
Molecular weight
Liquid density (kg/m3 )
Kinematic viscosity (m2 /s) at 40◦ C
Lower heating value, mass (MJ/kgfuel )
Lower heating value, volume (MJ/Lfuel )
Boiling point at 1 bar (◦ C)
Vapor pressure at 38◦ C (bar)
Cetane number
Stoichiometric A/F ratio
Diesel (2D)
Dimethyl ether
(DME)
C15 H25
170--200
820--860
2.8 ×10−6
42.5
34.8--36.5
180--360
0.0069
40--55
14.7
CH3 OCH3
46.07
668
2.2 ×10−7
28.4
19.0
−25
8
55--60
9.0
Rapeseed methyl
ester (RME)
882
4.1×10−6
37.7
48--58
11.2--12.5
Diesel Fuels
285
where
𝐺 = API gravity = 141.5 /(Specific gravity) −131.5
𝑇50 = Midpoint boiling temperature, ◦ F
The cetane index is useful because it is less expensive to obtain than a measurement of
the actual cetane number. It is also used as a surrogate specification of aromatics content.
In addition, it illustrates that not all diesel fuel properties can be specified independently
of one another.
The regulated emissions from vehicular diesel combustion include CO, HC, NO𝑥 ,
and particulate matter (PM). The emissions regulations have been imposed in response
to concerns about the adverse effect that compression ignition engines have had on ambient air quality, specifically NO𝑥 and PM. Nitrogen oxides are a precursor to groundlevel ozone formation, and particulate emissions are a respiratory hazard. As discussed
earlier, there is a trade-off between NO𝑥 and particulate matter (PM) emissions from
compression ignition engines, as techniques to lower NO𝑥 will generally increase PM,
and vice versa. With the increased availability of diesel after treatment devices, such as
lean NO𝑥 traps, U.S. manufacturers have increased their production and sales of diesel
vehicles.
9.8.1 Alternative Fuels for Compression Ignition Engines
A number of fuels can be used as an alternative for petroleum-based diesel fuels. One
alternative is biodiesel, that is, methyl ester vegetable oils, such as soybean, castor, canola,
sunflower, cotton, palm, coconut, Jatropha, and algae oils. Other diesel alternative fuels
are dimethyl ether (DME), and Fischer--Tropsch (F--T) fuel.
Biodiesel fuels are designated with the prefix B, so a mixture of 20% biodiesel is
labeled B20. A number of U.S. states have mandated the use of biodiesel, and the levels
vary by state, from B2 to B20. Diesel engines are rated for a maximum percentage of
biodiesel, typically from B5 to B20. The most common blend in the United States is B20.
In Europe, diesel fuel is blended with 7% biodiesel to produce B7. Biodiesel is also a ULSD
(Ultralow sulfur diesel) because it contains very low levels of sulfur.
Biodiesel fuels are produced through the transesterification of triglycerides in vegetable
oil using a low molecular weight alcohol, such as methanol. The methyl ester is obtained
through a process in which methyl alcohol and a catalyst (such as sodium hydroxide or
potassium hydroxide) chemically breaks down the triglyceride molecule into methyl esters
of the oil and a glycerin byproduct. Biodiesel is not a single chemical compound, as the
triglycerides in vegetable oils are a variable mixture of unsaturated and saturated fatty
acids.
The thermodynamic properties of a widely used biodiesel fuel, rapeseed methyl ester
(RME) are listed in Table 9.18. The interest in algae-based biodiesel has been increasing,
due to the far greater sunlight--oil conversion efficiency of algae relative to land-based
crops.
Fischer--Tropsch (F--T) fuel is produced from a mixture of CO and H2 using a catalytic
reforming process with iron or cobalt. The CO is generated by pyrolysis of woody materials,
such as switchgrass. The Fischer--Tropsch reaction is a water--gas reaction:
𝑛CO + (2𝑛 + 1)H2 → C𝑛 H2𝑛+2 + 𝑛H2 O
(9.6)
286
Fuels
Dimethyl ether (DME) is an oxygenated fuel produced by dehydration of methanol or
from synthesis gas. The volumetric energy density (MJ/L) of DME is about half that of
diesel fuel. It burns with a visible blue flame, similar to that of natural gas. It is noncorrosive
to metals but does deteriorate some elastomers.
Alternative diesel fuels have a higher cost, and lower volumetric energy density than
fossil-based diesel fuel, but do produce lower CO and particulate emissions. Numerous
studies have shown slightly greater NO𝑥 levels from diesel engines fueled with biodiesel
relative to petroleum-based diesel. For example, Krahl et al. (1996) report that RME had
about 40% lower HC emissions, 35% lower CO, 35% lower PM, but about 15% greater
NO𝑥 emissions. One explanation is that the oxygen atoms in the biodiesel molecule give
rise to a leaner premixed ignition zone during combustion resulting in increased combustion
temperatures.
Clark et al. (1999) measured the transient emissions of a number of blends of Fischer-Tropsch fuel. All of the regulated emissions were lower in comparison with low-sulfur
diesel fuel, with 43% lower HC emissions, 39% lower CO, 14% lower PM, and 14% lower
NO𝑥 emissions.
9.9 REFERENCES
BASS, E., B. BAILEY, and S. JAEGER (1993), ‘‘LPG Conversion and HC Emissions Speciation of a
Light Duty Vehicle,’’ SAE paper 932745.
BLACK, F. (1991), ‘‘An Overview of the Technical Implications of Methanol and Ethanol as Highway
Motor Vehicle Fuels,’’ SAE paper 912413.
CADLE, S., P. GROBLICKI, R. GORSE, J. HOOD, D. KARDUBA-SAWICKY, and M. SHERMAN (1997), ‘‘A
Dynamometer Study of Off-Cycle Exhaust Emissions--The Auto/Oil Air Quality Improvement
Research Program,’’ SAE paper 971655.
CLARK, N., C. ATKINSON, G. THOMPSON, and R. NINE (1999), ‘‘Transient Emissions Comparisons of
Alternative Compression Ignition Fuels,’’ SAE paper 1999-01-1117.
DHALIWAL, B., N. YI, and D. CHECKEL (2000), ‘‘Emissions Effects of Alternative Fuels in Light-Duty
and Heavy-Duty Vehicles,’’ SAE paper 2000-01-0692.
FULTON, J., F. LYNCH, R. MARMARO, and B. WILLSON (1993), ‘‘Hydrogen for Reducing Emissions
from Alternative Fuel Vehicles,’’ SAE paper 931813.
HURN, R. W. and H. M. SMITH (1951), ‘‘Hydrocarbons in the Diesel Boiling Range,’’ J. Ind. Eng.
Chem., Vol. 43, pp. 2788--2793.
IKUMI, S. and C. WEN (1981), ‘‘Entropies of Coals and Reference States in Coal Gasification Availability Analysis,’’ West Virginia University Internal Report.
KATO, K., K. IGARASHI, M. MASUDA, K. OTSUBO, A. YASUDA, and K. TAKEDA (1999), ‘‘Development
of Engine for Natural Gas Vehicle,’’ SAE paper 1999-01-0574.
KRAHL, J., A. MUNACK, M. BAHADIR, L. SCHUMACHER, and N. ELSER (1996), ‘‘Review: Utilization of
Rapeseed Oil, Rapeseed Oil Methyl Ester or Diesel Fuel: Exhaust Gas Emissions and Estimation
of Environmental Effects,’’ SAE paper 962096.
KUKKONEN, C. and M. SHELEF (1994), ‘‘Hydrogen as an Alternative Fuel,’’ SAE paper 940766.
LOVELL, W. (1948), ‘‘Knocking Characteristics of Hydrocarbons," J. Ind. Eng. Chem., Vol. 40, pp.
2388--2438.
MALENSHEK, M. and D. OLSEN, (2009), ‘‘Methane number testing of alternative gaseous fuels’’, Fuel,
Vol. 88, p. 650--656.
MIDGLEY, T. and T. BOYD (1922), ‘‘The Chemical Control of Gaseous Detonation with Particular
Reference to the Internal-Combustion Engine,’’ J. Ind. Eng. Chem., Vol. 14, pp. 894--898.
OWEN, K. and T. COLEY (1995), Automotive Fuels Reference Book, Society of Automotive Engineers,
Warrendale, Pennsylvania.
Homework
287
SPRINGER, K., L. SMITH, and A. DICKINSON (1994), ‘‘Effect of CNG Start-Gasoline Run on Emissions
from a 34 Ton Pick Up Truck,’’ SAE paper 941916.
SUN, X., A. LUTZ, E. VERMIGLIO, M. AROLD, and T. WIEDMANN (1998), ‘‘The Development of the GM
2.2L CNG Bi-Fuel Passenger Cars,’’ SAE paper 982445.
WEBB, R. and P. DELMAS (1991), ‘‘New Perspectives on Auto Propane as a Mass-Scale Motor Vehicle
Fuel,’’ SAE paper 911667.
9.10 HOMEWORK
9.1
What is the chemical structure of (a) 3-methyl-3-ethylpentane, and (b) 2,4-diethylpentane?
9.2
If a hydrocarbon fuel is represented by the general formula C𝑥 H2x , what is its stoichiometric
mass air--fuel ratio?
9.3
A fuel has the following composition by mass: 10% pentane, 35% heptane, 30% octane,
and 25% dodecane. If its general formula is of the form C𝑥 H𝑦 , find 𝑥 and 𝑦.
9.4
If the mass composition of a hydrocarbon fuel mixture is 55% paraffins, 30% aromatics,
and 15% monoolefins, what is its specific heat? Compare with the value of specific heat
for C7 H17 using Equation 3.37. Assume 𝑇 = 1000 K.
9.5
Compute the enthalpy of formation of C8 H18 .
9.6
A four-stroke engine operates on methane with an equivalence ratio of 0.9. The air and fuel
enter the engine at 298 K, and the exhaust is at 800 K. The heat rejected to the coolant is
350 MJ/kmolfuel . (a) What is the enthalpy of the exhaust combustion products? (b) What
is the specific work output of the engine? (c) What is the first law efficiency of the engine?
9.7
A fuel blend has a density of 700 kg/m3 and a midpoint boiling temperature of 90◦ C. What
is its cetane index?
9.8
What is the decrease in volumetric efficiency of a 5 L gasoline-fueled engine when it is
retrofitted to operate with propane? The decrease is due to the displacement of a portion
of the inlet air by the propane fuel. Assume standard temperature and pressure inlet air
conditions.
9.9
Repeat Problem 9.8 for hydrogen and methane.
9.10
A flexible fuel vehicle operates with a mixture of 35% isooctane and 65% methanol, by
volume. If the combustion is to be stoichiometric, what should the mass air--fuel ratio be?
9.11
If an dragster fuel tank contains a mixture of 70% octane and 30% nitromethane, by volume,
what should the mass air--fuel ratio be to run rich at 𝜙 = 1.24?
9.12
A vehicle is equipped with a flex-fuel eight-cylinder 6.0 L spark ignition engine running
on a mixture of octane and ethanol at 𝜙 = 0.95. At an operating condition of 2500 rpm,
the thermal efficiency is 0.30 and the volumetric efficiency is 0.80. If the fuel mixture
is changed from 10% ethanol (E10) to 85% ethanol (E85), (a) What is the change in the
overall mass air--fuel ratio, and (b) What is the change in the engine power?
9.13
Verify the CO2 concentration values resulting from the combustion of propane, methane,
methanol, ethanol, and gasoline given in Table 9.6.
Chapter
10
Friction and Lubrication
10.1 INTRODUCTION
In this chapter, we will examine the frictional processes in internal combustion engines
and learn about the properties of engine lubricating oils. Engines are lubricated not only to
reduce friction but also to prevent engine failure. Since frictional losses are a significant
fraction of the power produced in an internal combustion engine, minimization of friction
has long been a major consideration in engine design and operation. The friction forces in
engines are a consequence of both hydrodynamic stresses in oil films and also metal-tometal contact. Minimum friction is obtained by use of low shear strength lubricating oil
and hardened metallic surfaces.
Lubricants also reduce component wear during metal-to-metal contact, and reduce corrosion from the acidic products of combustion. Additives are added to lubricating oils that
preferentially adsorb to bearing surfaces and lower the coefficient of friction. A pressurized
system is used to circulate lubricants to the bearings, piston rings, and valve trains.
The frictional processes in an internal combustion engine can be categorized into three
main components: (1) the mechanical friction, (2) the pumping work, and (3) the accessory
work. The mechanical friction includes the friction of internal moving parts such as the
crankshaft, piston, rings, and valve train. The pumping work is the net work done to draw
in a fresh mixture during the intake stroke and push out the combustion gases during the
exhaust stroke. The accessory work is the work required for operation of accessories such
as the oil pump, fuel pump, alternator, and a fan.
The study of friction in engines is highly empirical and depends on experimental
measurements, especially, as we will see, on motored engine tests. We will use scaling
arguments to develop relations for the dependence of the various modes of friction work
on overall engine parameters such as the bore, stroke, and engine speed, then construct an
overall engine friction model. The coefficients for the scaling relations are obtained from
experimental data and include lubrication oil properties such as viscosity.
10.2 FRICTION COEFFICIENT
The friction coefficient 𝑓 defined in Equation 10.1 is the ratio of the shear force 𝐹f to
the normal force 𝐹n acting on a surface. The friction coefficient depends on the type of
lubrication and the fluid stress between two surfaces.
𝐹
𝜏
(10.1)
𝑓= f =
𝐹n
𝜎n
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
288
Friction Coefficient
Boundary
289
Hydrodynamic
Mixed
Coefficient of friction (f)
100
Figure 10.1 Stribeck diagram showing
friction regimes.
10–1
10–2
10–3
10–4
Stribeck variable μN/P
Generally speaking, there are three friction regimes: hydrodynamic, mixed, and boundary friction. In an engine, the frictional losses have a variety of sources, for example,
hydrodynamic friction in the crankshaft and camshaft journal bearings, mixed lubrication
in the piston ring pack and skirt, and rolling contact friction in the valve train. The friction
mean effective pressure (fmep) of an engine component depends on the friction regime
and the lubricating surface geometry, such as whether the contacting surfaces are sliding
(piston) or rotating (bearings) relative to each other.
The three friction regimes are shown on a Stribeck diagram in Figure 10.1. The
diagram is named after Richard Stribeck (1861--1950), a German engineering professor
who published pioneering studies of rotating bearing friction in 1901. The Stribeck diagram
plots the friction coefficient as a function of a Stribeck variable or duty parameter. For
rotating surfaces, the duty parameter is 𝜇𝑁∕𝜎n , where 𝜇 is the lubricant dynamic viscosity,
𝑁 is the relative rotational speed between surfaces, and 𝜎n is a normal stress, that is, the
fluid pressure. For sliding surfaces, the duty parameter is 𝜇𝑈 𝐿∕𝐹n , where 𝑈 is the relative
velocity of the two surfaces, and 𝐿 is the contact length in the direction of motion. In the
hydrodynamic friction regime, the surfaces are completely separated by a liquid film of
thickness ℎ. This is a preferred mode of operation since mechanical wear is minimized. The
shear stress is entirely due to the lubricant viscosity. Therefore in this regime, the friction
coefficient 𝑓 is given by Equation 10.2:
𝑓=
𝜇 𝑑𝑢
𝜎n 𝑑𝑦
(10.2)
and is a straight line on the Stribeck diagram. Crankshafts, connecting rods, and piston
rings are designed to operate in the hydrodynamic regime as much as possible. An increase
in lubricant temperature will decrease its viscosity, decreasing the duty parameter and the
coefficient of friction. The relationship between the film thickness and the Stribeck duty
parameter is shown schematically in Figure 10.2, indicating that the film thickness increases
nonlinearly as the duty parameter increases.
As the pressure load on the lubricant is increased or the relative velocity decreased,
the oil film thins out to the point where its thickness is comparable to the size of the surface
irregularities. This is the mixed lubrication regime. The liquid film no longer completely
separates the surfaces, and intermittent metal-to-metal contact occurs, causing an increase
in the friction coefficient. The friction coefficient in the mixed regime is a combination of
hydrodynamic and metal-to-metal contact friction.
Friction and Lubrication
Film thickness h
290
Figure 10.2 Schematic of lubricant
film thickness versus Stribeck duty
parameter.
Duty parameter μN/P
With further increase in load or decrease in speed, the metal-to-metal boundary lubrication regime is reached. Boundary lubrication occurs at either end of the piston stroke when
the piston velocity approaches zero, in relatively slow moving valve train components, and
during engine start-up and shutdown. In boundary lubrication, oil film patches separate the
sliding surfaces where the thickness is just a few molecular diameters of the lubricant, as
shown in Figure 10.3. The force required to cause tangential motion in boundary lubrication
is approximately the area of contact times the shear strength of the adsorbed oil layer, 𝜎o .
It is important to know how the lubricant can be adsorbed by the surfaces, how rough the
surfaces are, and whether or not the surface molecules themselves are prone to adhering
to one another. The real area of contact depends primarily on the applied load, the yield
strength, and the asperities of the softer material. The yield stress, 𝜎m , of the softer material
balances the applied load, so that as the load increases, there is a proportional increase in
the area of contact. The coefficient of friction in the metal-to-metal boundary lubrication
regime is given by Equation 10.3:
𝑓=
𝜎o
𝜎m
(10.3)
and is independent of the engine design and operating parameters such as engine speed. The
friction depends on the properties of the lubricant (excluding viscosity), and the properties
of the sliding surfaces, such as the roughness, plasticity, elasticity, shear strength, and
hardness.
Relative velocity
Adsorbed
oil film
Figure 10.3 Metal-to-metal contact in
boundary lubrication. Adapted from
Rosenberg (1982).
Contact area
Friction Measurements
291
10.3 FRICTION MEAN EFFECTIVE PRESSURE
The engine friction work, 𝑊f reduces the mechanical efficiency of an engine, and is the
difference between the indicated work 𝑊𝑖 and the brake work 𝑊b . The friction work is
eventually removed as waste heat by the engine cooling system. We can represent friction
as a power loss:
𝑊̇ f = 𝑊̇ 𝑖 − 𝑊̇ b
(10.4)
It is useful to normalize the engine friction power into a mean effective pressure (mep)
form by dividing by displacement volume and engine speed so that the friction losses of
engines of different sizes can be directly compared.
fmep =
𝑊̇ f
𝑉d 𝑁∕2
(10.5)
Therefore,
fmep = imep − bmep
(10.6)
The indicated mean effective pressure (imep) is the net work per unit displacement
volume done by the gas during compression and expansion, and the brake mean effective
pressure (bmep) is the external shaft work per unit displacement volume done by the
engine. With this definition, the pumping losses during the intake and exhaust strokes are
considered to be part of the overall engine friction. It is customary to also include the work
to run auxiliary components with the mechanical friction when the friction mean effective
pressure is defined as in Equation 10.6. Accordingly, the friction mean effective pressure is
the sum of the mechanical friction (mfmep), pumping (pmep), and accessory (amep) mean
effective pressure.
fmep = mfmep + pmep + amep
(10.7)
If a supercharger is connected to the engine crankshaft, then
fmep = imep − bmep − cmep
(10.8)
The term cmep is the work per unit volume required to power the supercharger compressor.
The friction power 𝑊̇ f is the product of the friction force 𝐹f and a characteristic
velocity 𝑈 :
𝑊̇ f = 𝐹f 𝑈
(10.9)
Therefore, the mechanical friction mean effective pressure scales as
mfmep ∼
𝐹f 𝑈
𝑊̇ f
∼
𝑉d 𝑁
𝑛c 𝑏2 𝑠𝑁
(10.10)
where 𝑉d is the displacement volume, 𝑛c is the number of cylinders, 𝑏 is the cylinder bore,
𝑠 is the piston stroke, and 𝑁 is the engine speed.
10.4 FRICTION MEASUREMENTS
There are a number of measurement methods used to determine the friction force and
the friction mean effective pressure. The most direct method is to use Equation 10.6.
The indicated mean effective pressure is computed from cylinder pressure measurements
during compression and expansion, and the brake mean effective pressure is determined
Friction and Lubrication
from dynamometer measurements. Such measurements are discussed in Chapter 11. A
more commonly used method that does not require the measurement of cylinder pressure
is motoring the engine without combustion. This method measures the motoring mean
effective pressure (mmep), defined as the work per unit displacement volume required to
rotate an engine operated without combustion. A direct current (dc) electric cradle-type
dynamometer is an appropriate apparatus for such a measurement. The dynamometer is
mounted on bearings and is restrained from rotation by only a strut connected to a load
beam. When the dynamometer is absorbing or providing power, a torque is applied to the
load cell of the dynamometer either by the engine or by the dc motor. The work done
in rotating the crankshaft through one revolution is 2𝜋𝜏. Hence, for two- and four-stroke
engines, respectively
mep =
4𝜋𝜏
𝑉d
(four-stroke engine)
(10.11)
mep =
2𝜋𝜏
𝑉d
(two-stroke engine)
(10.12)
The above equations are valid whether or not the engine is firing. Although the sign of
𝜏 changes, we think of bmep and mmep as positive numbers. Practical application comes
from the observation that under controlled conditions,
fmep ≃ mmep
(10.13)
Therefore, motoring tests are useful because it is very much easier to measure mmep
than fmep. However, it needs to be noted that the motored engine friction and the fired
engine friction are not the same, since the thermodynamic state of the engine is different
in each case. The motored friction can be more or less than the fired friction, depending on
the relative magnitudes of the various friction components. The piston and cylinder bore
temperatures are lower for the motored state, which will increase the viscosity and thus the
hydrodynamic friction of the lubricating film, but also increase the clearances that lowers
the friction. Since the combustion pressure on the piston rings is much greater for the fired
case, the ring friction for the fired case will be greater than the motored case. Similarly,
the bearing loads will be greater for the fired case. The cooler exhaust gas for the motoring
case will have a greater density, producing a larger motored pmep. Finally, temperature
gradients within the engine are much less in the motored engine than in the fired engine. To
obtain approximately the same heat loss, oil and coolant temperatures should be matched
between the motored and fired engines.
Comparisons between motoring and firing friction are given in Figure 10.4 for a diesel
engine, and Figure 10.6 for a spark ignition engine. The total fmep is on the order of 1
bar for both engines. The results illustrate the differences between a fired and a motored
test. In both figures, the friction for motored and fired cases is determined by integrating
Engine motored complete
Engine motored manifolds removed
+ Engine motored, valves removed, and
camshaft deactivated
2.0
Fmep (bar)
292
Figure 10.4 Diesel engine fmep
versus piston speed. (Brown,
1973.)
1.5
Engine
firing
1.0
0.5
0
4
5
6
7
8
9
10
Piston speed (m/s)
11
12
13
Friction Measurements
293
the pressure--volume diagram to obtain the imep and subtracting the measured bmep as
per Equation 10.6. Therefore, the motored friction labeled in Figures 10.4 and 10.6 is the
motored fmep, not the mmep, which is obtained from dynamometer measurements alone.
The diesel engine tested by Brown (1973) in Figure 10.4 is a six-cylinder, in-line
engine. As the piston speed is increased from 4 to 12 m/s, the fmep increases nearly
linearly, with no significant variation with load. The motored fmep is slightly greater than
the fired fmep, a consequence of cooler wall temperatures and higher oil viscosity during
motoring. Since the airflow in a diesel engine is unthrottled, the pumping friction is not a
function of the load.
The most important advantage of a motoring test is that the engine can be partially
disassembled, which precludes firing, and motored to study the distribution of friction
among the various parts. Figure 10.5 is typical of the results that can be obtained by
this method. In this experiment by Brown (1973), the diesel engine was systematically
disassembled, and the resulting fmep measured. To measure the pumping loss, the first
items removed were the manifolds and turbocharger. Next the valves were removed and
the camshafts disconnected. As disassembly proceeds, it became clear that pistons and
rings are responsible for about one-half to three fourths of the friction. The last curve, H,
shows the friction due to spinning only the crankshaft.
The spark ignition engine of Figure 10.6 is a four-cylinder, in-line engine with a
12 : 1 compression ratio. The mechanical friction (mfmep) and pumping (pmep) are plotted
versus load for a constant mean piston speed of 6.1 m/s. As the throttle of the spark
3.0
A
Mean effective pressure (bar)
2.5
B
2.0
C
1.5
D
E
F
G
1.0
0.5
H
0
5
6
7
8
9
10
Piston speed (m/s)
11
12
13
Engine setup
Figure 10.5 Motored friction mean
effective pressure (fmep) during
disassembly of a diesel engine.
Adapted from Brown (1973).
A Complete engine
B Complete engine minus intake and exhaust manifolds
C Setup B minus all valves, camshaft, and measured
pumping loss
D Setup C minus water pump
E Setup D minus oil pump
F Setup E minus all top and intermediate piston rings
G Setup E minus all piston rings
H Crankshaft only
294
Friction and Lubrication
2.0
Total
mep (bar)
1.5
Mechanical friction
1.0
Pumping
0.5
Firing
Motoring
Figure 10.6 Gasoline engine
friction versus load.
(Gish et al., 1957.)
0
1
2
3
4
5
6
7
Load, bmep (bar)
8
9
10
ignition engine is opened to meet an increased load, the pmep decreases for both the fired
and motored cases. However, the fired mechanical friction increases with load due to the
increased gas pressure, while the motored mechanical friction is unchanged. Consequently,
for this engine, the total fired friction is greater than the motored total friction. Note that
Equation 10.13 is a rather crude approximation at high loads for this spark ignition engine.
For further reading in this area, an overview of techniques for measuring friction in fired
engines is given in Noorman et al. (2000).
10.5 FRICTION MODELING
The above experimental results indicate that friction depends on the piston speed and the
load. Other variables that immediately come to mind are the oil viscosity 𝜇o , oil density
𝜌o , the compression ratio 𝑟, the bore 𝑏, and other geometrical parameters that specify the
bearings, rings, pistons, cams, lifters, and so on. Mathematically,
]
[
(10.14)
f mep = 𝑓 𝑈̄ p , 𝜇o , 𝜌o , 𝑟, imep, 𝑏, 𝑙𝑖 (𝑖 = 1, 2, ...), 𝜎𝑗 (𝑗 = 1, 2, ....)
where the 𝑙𝑖 (𝑖 = 1, 2, ...) represent all the lengths and 𝜎𝑗 (𝑗 = 1, 2, ....) all the relevant material
properties. The dimensionless groups are therefore
]
[
𝜌o 𝑈̄ p 𝑏 imep 𝑏 𝑙𝑖 𝜎j
f mep
, ,
= 𝑓1
, 𝑟,
(10.15)
𝜇o
𝜌o 𝑈̄ p2
𝜇o 𝑈̄ p 𝑏 𝑏
The first dimensionless group in the argument list is a Reynolds number based on the piston
speed, bore, and the oil dynamic viscosity 𝜇o .
𝑅𝑒 =
𝜌o 𝑈̄ p 𝑏
𝜇o
=
𝑈̄ p 𝑏
𝜈o
(10.16)
since 𝜈o = 𝜇o ∕𝜌o . In the case of motoring mean effective pressure, the dependence on load
is no longer necessary and we conclude that
]
[
𝑙 𝜎𝑗
mmep
= 𝑓2 𝑅𝑒, 𝑟, 𝑖 ,
(10.17)
𝑏 𝑏
𝜌o 𝑈̄ p2
Journal Bearing Friction
295
That the Reynolds number based on mean piston speed is the proper scaling parameter
has been demonstrated by Taylor (1985), who did experiments with geometrically similar
engines. Geometrically similar engines refer to any family of engines manufactured from
the same materials with same 𝑙𝑖 ∕𝑏, 𝜎𝑖 ∕𝑏, that is, same compression ratio and stroke to bore
ratio. In Taylor’s experiments, the motoring mean effective pressures for three different
engines (with 𝑏 = 6.35 cm, 10.2 cm, and 15.2 cm), fell on a common curve when plotted as
a function of mean piston speed. The oil viscosities were chosen to be proportional to the
bore, so the mmep was a function of the Reynolds number. This implies that
mmep
= 𝑓3 (𝑅𝑒)
(Geometrically similar engines)
(10.18)
𝜌o 𝑈̄ 2
p
When one recognizes that to a certain degree, all reciprocating piston engines are geometrically similar, then Equation 10.18 applies in an order of magnitude analysis for any engine.
Since the viscosity of oil used in engines is more or less independent of size, the friction
can be expected to be relatively less in larger engines than in smaller engines.
As an illustration of the application of Equation 10.18, the correlation of Bishop (1964)
is
mmep 1.3 × 105 1.7 × 108
(𝑟 + 15)
=
+
𝑅𝑒
𝑅𝑒2
𝜌o 𝑈̄ p2
(10.19)
Equation 10.19 implies that the motoring mean effective pressure decreases as engine
size increases, as shown by writing it in dimensional form:
mmep = 1.3 × 105
𝑈̄ p 𝜇
𝑏
+ 1.7 × 108 (𝑟 + 15)
𝜇2
𝜌𝑏2
(10.20)
Engine friction models have been developed to account for different types of friction
and are used in the engine design process to estimate the influence of parameters such
as engine speed and geometry on engine friction. The process used in the modeling is
to develop component fmep models, then sum the component fmep values to arrive at a
prediction of overall engine fmep. Papers by Sandoval and Heywood (2003) and Shayler
et al. (2005) develop engine friction models and explicitly include the dependence on oil
viscosity. There has been a continued decease in engine friction over the last 50 years due
to advances in component design and manufacturing, so the coefficients used in friction
models always need to be adjusted for a given engine.
10.6 JOURNAL BEARING FRICTION
The journal bearings used in an internal combustion engine include the main crankshaft
bearings, connecting rod bearings, and accessory bearings and seals. A crankshaft with main
and connecting rod journal bearing shafts is shown in Figure 10.7. Note the lubrication
ports on the shafts and the different main and connecting rod bearing diameters. A journal
bearing is shown schematically in Figure 10.8. The journal bearing is cylindrical, with
as smooth a finish as possible. The journal bearing operates in the hydrodynamic friction
regime during normal operation. During start-up and shutdown, the friction regimes are
mixed and boundary friction due to the low bearing speed.
Journal bearings are relatively soft to allow foreign particles to be embedded without
damaging the journal and have a low melting point to reduce the risk of seizure. Materials
used in journal bearings in internal combustion engines are composed of babbit, which is an
alloy of lead (Pb), tin (Sn), antimony (Sb), and copper (Cu), specified by ASTM Standard
296
Friction and Lubrication
Figure 10.7 Engine crankshaft.
(Courtesy Norton Manufacturing.)
N
e
R
x
Figure 10.8
Journal bearing geometry.
B23-2010. The bearing material is named after Isaac Babbit (1799--1862), an American
engineer who invented a low-friction tin-based metal alloy for use in bearings. The babbit
alloy has a relatively low load-carrying capacity, so it is bonded to a stronger substrate
such as steel or aluminum. Babbit thicknesses in engine bearings currently range from 20
to 200 µm, a significant improvement from typical babbit thicknesses of the order of 5 mm
in 1900. A lead-based babbit is composed of roughly 89% Pb, 9% Sn, and 2% Cu.
Loads on crankshaft bearings vary significantly with crank angle, the connecting rod
geometry, and the combustion gas pressure. To meet the bearing loads, crankshaft main
bearings for automotive spark ignition engines are typically sized to be about 65% of the
cylinder bore, and connecting rod bearings are sized to be about 55% of the cylinder bore.
Bearing lengths are sized at 35--40% of the cylinder bore. In addition to the crankshaft
and connecting rod journal bearings, there are journal bearings on the camshaft rotating at
one-half the engine speed. Also, the piston pin on the connecting rod oscillates back and
forth without completing a revolution.
The difference in the diameters of the inner shaft and the outer bearing creates a thin
annulus through which the lubricant flows. At rest the shaft sits in contact with the bottom
of the bearing. As the shaft begins to rotate, its center line shifts eccentrically in the bearing
maintaining metal-to-metal contact with the bearing. As the engine speed increases, the
bearing will ‘‘aquaplane’’ and enter the hydrodynamic regime, in which the rotating shaft
carries the oil all the way around the annulus between the two cylinders. If the bearing is
not sealed, oil will leak out at the ends, so oil is pumped at relatively low pressures through
internal passages to the bearing annulus.
Journal Bearing Friction
297
A minimum oil film thickness is required to maintain hydrodynamic lubrication as operation in the mixed or boundary lubrication regime increases engine wear. For automotive
class engines, the minimum thickness is on the order of 2µm. The oil film thickness can be
determined by resistance and capacitance measurements. Measurements by Tseregounis
et al. (1998) in a multicylinder engine indicated that the minimum film thickness occurs
during the power stroke of the cylinder nearest the bearing.
The oil film in a journal bearing is thin compared with the diameter of the bearing, so if
one neglects the bearing curvature, the flow in a journal bearing can be modeled as Couette
flow with constant dynamic viscosity 𝜇. The velocity gradient in a bearing of diameter 𝐷b ,
length 𝐿b , and annular clearance 𝑐 is therefore
𝑑𝑢
= 𝜋𝐷b 𝑁∕𝑐
𝑑𝑟
(10.21)
The friction force, 𝐹f , in the bearing is
𝐹f = 𝐴𝜇
𝑑𝑢
= (𝜋𝐷b 𝐿b )𝜇𝜋𝐷b 𝑁∕𝑐 = 𝜋 2 𝜇𝐷b2 𝐿b 𝑁∕𝑐
𝑑𝑟
(10.22)
Equation 10.22 is known as Petrov’s equation, named after the Russian scientist Nikolai
Petrov (1836--1920), who published his analyses of hydrodynamic flow in bearings in 1883.
The friction coefficient in the Petrov model is
𝑓=
𝐹f
= 𝜋(𝜇𝑁∕𝑃 ) 𝐷b ∕𝑐
𝑃 𝜋𝐷b 𝐿b
(10.23)
In the hydrodynamic regime, the Petrov flow model predicts that the friction coefficient
increases linearly with the Stribeck variable, and the slope is dependent on the ratio of the
bearing diameter to clearance. The friction of real bearings approaches the Petrov value at
high values of the Stribeck variable, 𝜇𝑁∕𝑃 . More rigorous analyses include the bearing
load and eccentricity, but the results still retain the scaling of the Petrov equation.
Using the Petrov equation, the nondimensional fmep of 𝑛c journal bearings is inversely
proportional to the oil Reynolds number:
⎡ 𝐷 b 𝐿b ⎤
f mep
1 ⎢ 𝑏 𝑠 ⎥ 1
∼
𝑛c ⎢ 𝑐 ⎥ 𝑅𝑒
𝜌o 𝑈̄ p2
⎣ 𝐷b ⎦
(10.24)
The friction mean effective pressure of a journal bearing array, such as the crankshaft
main bearings or the connecting rod bearings, scales with engine speed to the 0.6 power
(Shayler et al. 2005), assuming constant bearing clearance and oil viscosity, as shown in
Equation 10.25:
f mepbearings ∼
𝐹f 𝑈
𝑛c 𝑁𝑏2 𝑠
)(
(
)0.4
𝑛b 𝑁 0.6 𝐷b3 𝐿b
𝜇
= 𝑐b
𝜇ref
𝑛c 𝑏2 𝑠
(10.25)
Shayler et al. (2005) suggest a proportionality constant 𝑐b = 0.0202 (kPa-min0.6 /rev0.6 -mm)
for automotive diesel engines.
The crankshaft bearing seals operate in a boundary lubrication regime, since the seals
directly contact the crankshaft surface. As the normal force, which is the seal lip load, is
constant, the friction force will be constant, and the friction mean effective pressure of the
298
Friction and Lubrication
crankshaft bearing seal will be independent of engine speed, and will scale as
fmepseals ∼
𝑁𝐷b
𝑛c
𝑁𝑏2 𝑠
= 𝑐s
𝐷b
𝑛c 𝑏2 𝑠
(10.26)
Shayler et al. (2005) suggest a proportionality constant 𝑐s = 9.36 × 104 kPa-mm2 .
10.7 PISTON AND RING FRICTION
Piston assembly friction is the major component of engine friction. The piston assembly is
composed of compression rings, oil control rings, piston skirts, and piston pins. Advances in
materials and increased understanding of friction have reduced the energy lost to friction;
however, due to the reciprocating relative motion between the cylinder and the piston
surfaces, piston and ring friction remain a major source of friction loss. Over the years,
due to improvements in materials and manufacturing techniques, piston skirts have become
smaller, the number of rings has decreased, and the piston has become lighter.
Illustrations of a piston and connecting rod, piston head and skirt, and a piston ring
assembly are shown in Figures 10.9--10.11, respectively, showing the relative sizes and
locations of the piston head, skirt, and rings.
Figure 10.9 Piston and
connecting rod. (Courtesy
Mahle, Inc.)
Piston and Ring Friction
299
Figure 10.10 Piston head for a
spark ignition engine. (Courtesy
Mahle, Inc.)
Upper compression
ring
Lower compression
ring
Oil control
ring
Figure 10.11 Piston ring assembly schematic.
Adapted from Merrion (1994).
The friction of the piston and rings results from the reciprocating contact between the
piston skirt and the ring pack with the cylinder bore. The cylinder bore is rougher than a
journal bearing bore, since the cylinder bore must retain some oil during operation. The
clearance 𝑐 between the piston and the cylinder wall is typically specified to be about
0.050 mm for street automobile engines and is increased to 0.100 mm or greater for race
engines to reduce friction losses.
The piston ring pack has three main functions. The rings seal the combustion chamber,
control the lubrication oil flow, and transfer heat from the piston to the cylinder. In order
to preserve a seal against the cylinder bore, each ring has some amount of radial tension.
Current ring pack designs generally use three piston rings, two compression rings, and an
oil control ring. Common types of piston rings are shown in Figure 10.12. Various cross
sections are available, such as rectangular, crown or barrel face, taper, and dykes. The top
compression ring can have a bevel on the upper inside edge of the ring to produce a positive
twist, and a seal on the bottom edge of the ring. The second ring can have a bevel on the
bottom inside edge to produce a reverse twist that will help scrape oil off the cylinder wall.
The oil control ring typically has two narrow rails and an expander that wipes off excess
oil from the cylinder liner.
Ring materials include cast iron, ductile (nodular) iron, and stainless steel. A ring pack
will often have a ductile iron top ring, a cast iron second ring, and a stainless steel oil
ring. Coatings available for ring faces are molybdenum and chrome. The space between
300
Friction and Lubrication
Plain
compression
Crown face
(barrel face)
Keystone
Taper face
(a) Upper and lower compression rings
Bevel
scraper
external
stepped
Bevelled
undercut
Combined
spacer
expander
Figure 10.12
piston rings.
Separate
spacer
expander
Common types of
(b) Oil control rings
the side of the ring and the groove is called the side clearance, and the space between
the back of the ring and the groove is called the back clearance. The back clearance is
minimized in high-performance engines to increase the sealing effectiveness. The side and
back clearances allow the pressure in the piston groove to follow the cylinder pressure, so
as the cylinder pressure increases, the increasing groove pressure presses the compression
rings more firmly against the cylinder wall, increasing the sealing. Piston rings are split,
forming an end gap, so that they can be slipped onto the piston and also accommodate
thermal expansion of the piston ring. Engine power is sensitive to the size of the end gap,
so the end gap is minimized in high-performance engines.
The piston skirt is designed to meet the side thrust forces originating from the rotation
of the connecting rod. The side thrust force on the piston skirt depends on the crank angle,
cylinder pressure, piston speed, acceleration, and connecting rod geometry. In addition, the
wrist pins are offset slightly by 1--2 mm to reduce the side thrust force. Offsetting the wrist
pins will also reduce piston noise.
Figure 10.13 illustrates a force balance applied to the piston. The forces on the piston
result from the gas pressure 𝑃 , connecting rod applied force 𝐹r , friction force 𝐹f , the
side thrust force 𝐹t , and the piston inertia 𝑚p 𝑎p . These forces can be resolved into 𝑥- and
𝑦- direction force balances as given by Equations 10.27 and 10.28:
∑
𝜋
𝐹𝑥 = 𝑚p 𝑎p = −𝐹r cos 𝜙 + 𝑃 𝑏2 ± 𝐹f
4
∑
𝐹y = 𝐹t − 𝐹r sin 𝜙 = 0
(10.27)
(10.28)
Piston and Ring Friction
301
y
Gas
pressure (P)
x
Side thrust (Ft)
b
Friction force (Ff)
Rod applied
force (Fr)
Figure 10.13
Piston force balance.
The sign on the friction force depends on the crank angle 𝜃. It is negative when the piston
is moving downward toward the crankshaft (0◦ < 𝜃 <180◦ ) and positive when the piston is
moving upward (180◦ < 𝜃 <360◦ ). Solving for the side thrust force 𝐹t :
)
(
𝜋
𝐹t = +𝑃 𝑏2 ± 𝐹f − 𝑚p 𝑎p
(10.29)
4
The piston acceleration 𝑎p , Equation 1.33 from Chapter 1, for small 𝜖 = 𝑠∕2𝑙 is
𝜔2 𝑠
(10.30)
[cos 𝜔𝑡 + 𝜖 cos 2𝜔𝑡]
2
Results of piston force balance calculations for a particular engine are given in Figure 10.14. A detailed calculation of side thrust force versus crank angle requires data from
the P--V diagram, the mass and moment of inertia of the piston and connecting rod, and
the engine speed. The largest side thrust forces occur during the expansion stroke when
𝑎p =
4
3
Full load 3800 rpm
Half load 2000 rpm
Side thrust (kN)
2
Figure 10.14 Piston side thrust
load and the switch of contact
sides. Adapted from Ting and
Mayer (1974).
1
200
400
600
0
Crank angle
(deg)
–1
180
–2
–3
360
540
720
Friction and Lubrication
Frictional force
(bar)
Piston area
0.4
0.3
fmep = 0.22 bar
0.2
Intake
0.1
0
–0.1
Compression
–0.2
0.5
0.4
Frictional force
(bar)
Piston area
302
fmep = 0.33 bar
Power
0.3
0.2
0.1
0
–0.1
Exhaust
–0.2
0
1
2
3
4
5
6
7
8
Piston travel (cm)
9
10
11
12
Figure 10.15 Piston and ring friction, 𝑈̄ p = 4.57 m/s, bmep = 5.78 bar, 𝑇c = 𝑇oil = 356 K (Leary
and Jovellanos, 1944).
the cylinder pressures are largest. The side thrust force changes sides as the piston passes
through top and bottom center since the connecting rod changes sides.
As a consequence of the force balance, the ‘‘left’’ side of the clockwise rotating piston
in Figure 10.13 is subjected to larger forces than the ‘‘right’’ side. In this case, the left
side is referred to as the major thrust side, whereas the right side is called the minor thrust
side. Since the friction work is the product of the friction force and piston velocity, the
friction work will be the largest during the middle of the stroke where the piston velocity
is greatest. Engines show more upper cylinder wear on the major thrust side than on the
minor thrust side as a result.
The friction due to the piston--ring assembly has been directly measured in a friction
research engine. Typical results are shown in Figure 10.15. The following features should
be noted (Taylor, 1985):
• The fmep due to piston and ring friction is on the order of 20--30 kPa for this engine.
• Friction forces occurring during expansion are about twice as large as those occurring
during any other stroke.
• Friction forces are comparable on the compression and exhaust strokes.
• Friction forces tend to be high just after top and bottom dead center, which Taylor
hypothesized was due to metallic contact between the rings and the cylinder wall.
• The friction force is nonzero at the top and bottom dead centers, due to the type of friction
research engine, which had a spring-loaded cylinder head for measurement of the axial
friction force.
Piston and Ring Friction
303
Support for Taylor’s hypothesis that metallic contact occurs in the vicinity of top
dead center comes from measurements of the oil film in running engines. Measurements
of the oil film thickness have been performed (e.g., see McGeehan, 1978) using electrical
resistance and capacitance techniques. The results show that resistance is low in the vicinity
of top and bottom dead center, as one would expect, if there is metallic contact. The
electrical resistance is higher at the middle of each stroke where piston speed is high and
hydrodynamic lubrication is expected. Similar results have been reported by Arcoumanis
et al. (1998) using a laser-induced fluorescence system. Therefore, it has generally been
concluded that a boundary lubrication regime exists at the ends of the stroke where the
piston speed is low, and hydrodynamic lubrication exists in the middle of each stroke where
the piston speed is higher.
Friction correlations for piston and ring friction have been developed that take both
the boundary lubrication and the hydrodynamic friction regimes into account. The hydrodynamic friction component depends on the contact area. Setting 𝐿s as the average piston
skirt and ring contact length, and 𝑐 as an average skirt clearance, the friction force, 𝐹f , of
the piston skirt scales as
𝑈̄ p
𝑑𝑢
∼ 𝑏𝐿s 𝜇
(10.31)
𝐹f = 𝐴𝜇
𝑑𝑦
𝑐
and for 𝑛c pistons, the piston skirt fmep scales as
fmepskirt ∼
𝐹f 𝑈
𝑛c 𝑁𝑏2 𝑠
=𝜇
𝐿s 𝑈̄ p
∼
(𝜇𝑛c 𝑏𝐿s 𝑈̄ p )𝑈̄ p
𝑐𝑛c 𝑁𝑏2 𝑠
(10.32)
𝑏𝑐
It is reasonable to assume that the piston skirt length and the clearance scale directly with the
bore, that is, 𝐿s ∼ 𝑏, and 𝑐 ∼ 𝑏. The skirt length scaling is based on geometrical similarity,
and the clearance scaling is based on thermal expansion considerations. Therefore, the
piston skirt hydrodynamic fmep is inversely proportional to the bore and can be expressed as
𝑈̄ p
𝑈̄ p
= 𝑐ps
(10.33)
fmepskirt ∼ 𝜇
𝑏
𝑏
Patton et al. (1989) suggest a coefficient 𝑐ps = 294 kPa-mm-s/m for the piston skirt friction,
including the oil properties in the proportionality constant.
The friction force of the piston rings has two components, one resulting from the ring
tension and the other component from the gas pressure loading. The component of piston
friction due to ring tension in the mixed lubrication regime will have a friction coefficient
inversely proportional to the engine speed. Patton et al. (1989) recommend a scaling
for piston--ring friction that bridges the boundary and hydrodynamic lubrication regimes
given by
)
(
1000
(10.34)
𝐹f ∼ 𝑓 𝐹n ∼ 1 +
𝑁
The piston ring scaling is
)
(
𝑛c 1 + 1000
𝑈̄ p
)
(
𝑁
1000 1
fmeprings ∼
= 𝑐pr 1 +
(10.35)
𝑁
𝑛c 𝑁𝑏2 𝑠
𝑏2
The proportionality constant recommended by Patton et al. (1989) is 𝑐pr =
4.06 × 104 kPa-mm2 .
Friction and Lubrication
100
90
80
70
Fmep (kPa)
304
60
ure
ress
50
p
Gas
40
gs
Rin
30
Skirt
20
10
Figure 10.16 Components of
piston friction.
0
1000
2000
3000
4000
Engine speed (rpm)
5000
6000
A correlation for the component of piston friction due to the gas pressure loading
recommended by Bishop (1964) is
)
𝑃 (
̄
(10.36)
fmepgas = 𝑐g 𝑖 0.088𝑟 + 0.182𝑟(1.33−𝐾 𝑈p )
𝑃a
where 𝑃𝑖 is the intake manifold pressure, 𝑃a is the atmospheric pressure, 𝑟 is the compression ratio, 𝑐g = 6.89, and 𝐾 = 2.38 × 10−2 s/m. The correlation includes the effect of
compression ratio and a decrease in the friction coefficient in the mixed lubrication regime.
Relative magnitudes of the three piston friction terms: skirt, ring pack, and bearings
are shown in Figure 10.16 for the 82-mm bore diesel engine specified in Table 10.4. The
skirt and rod bearing fmep increase linearly with engine speed, while the piston ring fmep
decreases with engine speed. At low speeds, most of the friction is due to the piston rings,
and at higher speeds, the majority of the friction is from the piston skirt.
Figure 10.17 illustrates some concepts used for a theoretical analysis of ring friction.
There is an oil layer separating the ring from the cylinder wall whose thickness, 𝛿, is both
time and spatially dependent. The oil pressure 𝑃oil is also time and spatially dependent.
Since the bore is much larger than the oil film thickness, a one-dimensional approximation
can be used. In the situation shown, the coordinate system is defined so that the ring is
stationary and the cylinder wall is moving with the instantaneous piston speed, 𝑈p . For the
case in which the oil film is much thinner than the ring width, that is
𝛿(𝑥, 𝑡) << 𝐿
(10.37)
the Navier--Stokes equations for the oil motion reduce to a Reynolds equation, which is
[
]
𝑑𝑃
𝑑𝛿
𝑑𝛿
𝑑
+ 12𝜇
(10.38)
𝛿 3 oil = 6𝜇𝑈
𝑑𝑥
𝑑𝑥
𝑑𝑥
𝑑𝑡
The boundary conditions to be applied to the Reynolds equation in the case illustrated by
Figure 10.17 are
𝑃oil (0, 𝑡) = 𝑃top (𝑡)
(10.39)
𝑃oil (𝐿, 𝑡) = 𝑃bot (𝑡)
(10.40)
305
Piston and Ring Friction
Hydrodynamic lubrication
Ptop(t)
c
y
0
(x, t)
Cylinder
wall
Ring
L
Oil
layer
Figure 10.17 Essential features of
a hydrodynamic analysis of ring
friction.
Piston
x
Up(t)
Instantaneous
piston speed
Pbot(t)
where 𝑃top (𝑡) and 𝑃bot (𝑡) are periodic functions known either from direct measurements
or from a combustion model complete with a blowby model. Numerical solution of the
Reynolds equation yields the oil film thickness 𝛿(𝑥, 𝑡) and the oil pressure 𝑃oil (𝑥, 𝑡) such
that the boundary conditions are satisfied.
Models that include both boundary and hydrodynamic lubrication are available for
analysis of the piston ring pack. The models also account for piston tilt, blowby and interring pressures, ring and piston curvature, ring tension, and surface roughness. The results
of the modeling predict the oil film thickness and pressure distribution for each ring. Some
calculated oil film thicknesses for a particular engine are shown in Figure 10.18. In the
Oil film thickness (microns)
10.00
2.50
1.00
0.25
bmep
(bar)
0
2.76
1.50
0.10
Power
0
Figure 10.18
Exhaust
180
N
(rpm)
1500
1500
1500
Inlet
360
Crank angle
Compression
540
720
Effect of load and speed on minimum oil film thickness (Allen et al., 1976).
306
Friction and Lubrication
Table 10.1 Types of Valve Trains
Type I
Type II
Type III
Type V
OHC
OHC
OHC
CIB
Direct acting/flat or roller follower
End pivot rocker/flat or roller follower
Center pivot rocker/flat or roller follower
Rocker arm/flat or roller follower
graph, the minimum oil thickness is plotted for different speeds and loads. The results are
similar in shape for all conditions and are fairly insensitive to speed and load. Surface
roughness in engines is such that for film thicknesses less than about a micron, metal-tometal contact can be expected to occur. The results in Figure 10.18 show that metal-to-metal
contact can be expected to occur at top and bottom dead center for all speeds and loads.
The results also show that at high load, metal-to-metal contact can occur for most, if not
all, of the power stroke.
10.8 VALVE TRAIN FRICTION
The valve train friction results from the camshaft, cam follower, and valve components.
Common valve train designs are listed in Table 10.1. The designs listed in Table 10.1
include overhead cam (OHC) and cam-in-block (CIB) with push rods. Either flat (ff) or
roller (rf) cam followers are used. A schematic of these valve train designs is given in
Figure 10.19. The shape and orientation of the camshaft lobes of a type V push rod V8
engine is shown in Figure 10.20. A type V rocker arm with a roller follower is shown
in Figure 10.21 and rocker arms mounted on a cylinder head are shown in Figure 10.22.
Example engine poppet valves are shown in Figure 10.23.
Type II
OHC end pivot rocker
Type V
Push rod
Sensitivity to speed, net torque
mechanical valve gear, 65 °C
Type III
OHC center pivot rocker
Type I
OHC direct active
Cam torque (Nm)
10
8
6
4
2
0
Figure 10.19
Various valve train designs (Rosenberg, 1982).
1000
3000
5000
Engine speed (rpm)
Valve Train Friction
Figure 10.20 Example V8
engine camshaft. (Courtesy
COMP Cams.)
Figure 10.21 Type V push rod
rocker arm with roller follower.
(Courtesy Jesel Valve train
Components.)
Figure 10.22 Rocker arms on
cylinder head. (Courtesy Jesel
Valve train Components.)
Figure 10.23 Engine poppet
valves. (Courtesy Wesco Valve.)
307
308
Friction and Lubrication
The valve train frictional losses are due to the following: hydrodynamic friction in
the camshaft bearing, mixed lubrication in the flat followers, rolling contact friction in the
roller followers, and both mixed and hydrodynamic friction due to the oscillating motion
of the lifters and valves.
The hydrodynamic friction in the camshaft is similar to that in the main and connecting
rod bearings. If the camshaft bearing diameter and length are assumed to be constant and
not a function of engine size, the fmep scales as
fmepcam ∼
𝑁𝑛cs
𝑛c
𝑏2 𝑠
= 𝑐c
𝑁𝑛cs
𝑛c 𝑏2 𝑠
(10.41)
where 𝑛cs is the number of camshaft bearings, assumed equal to the product of the number
of camshafts and the number of main bearings. Shayler et al. (2005) suggest a value of
𝑐c = 6720 kPa-mm3 -min0.6 /rev0.6 as the proportionality constant, plus an additional value
of 1.2 kPa to account for the camshaft oil seals.
For scaling purposes, the normal force in the valve mechanism components such
as followers, rocker arms, valve lifters, and valves is assumed to be proportional to the
product of the effective valve train mass and acceleration. The valve train effective mass
is proportional to valve area, which is in turn proportional to the cylinder cross-sectional
area. Therefore, the normal force in the valve train scales as the square of the cylinder bore.
A flat follower (ff) is assumed to operate in the mixed lubrication regime, and can be
scaled with a friction coefficient inversely proportional to engine speed.
(
)
10
(10.42)
𝑏2
𝐹f = 𝑓 𝐹n = 2 +
5 + 𝜇𝑁
Therefore, the flat follower fmep can be expressed as
)
(
𝑛v
𝐹f 𝑈
10
fmepff ∼
= 𝑐f f 2 +
5 + 𝜇𝑁 𝑛c 𝑠
𝑛c 𝑁𝑏2 𝑠
(10.43)
where 𝑛v is the total number of valves, and 𝑐f f is the flat follower coefficient.
A roller follower (rf) operates in the rolling contact friction regime and therefore is
scaled with a friction coefficient proportional to engine speed:
𝐹f = 𝑓 𝐹n ∼ 𝑁𝑏2
(10.44)
𝑛 𝑁
fmeprf = 𝑐rf v
𝑛c 𝑠
(10.45)
where 𝑐rf is the roller follower coefficient.
The coefficients for the valve train friction terms are given in Table 10.2 for the four
types of valve train mechanisms. In Table 10.2, the coefficients for the flat follower, the
roller follower, oscillating hydrodynamic, and oscillating mixed fmep have the units of
kPa-mm, kPa-mm-min/rev, kPa-(mm-min/rev)1∕2 , and kPa, respectively.
10.9 ACCESSORY FRICTION
The accessory mean effective pressure (amep) is the sum of the remaining crankcase friction
terms other than the journal bearing, piston and rings, and valve train losses. It includes the
oil pump (Figure 10.24), water pump (Figure 10.25), and noncharging alternator friction.
If these terms are assumed to be proportional to engine displacement, then by reference to
Equation 10.10, the mean effective pressure of the accessories is a function of engine speed
only and can be represented by a second-order polynomial.
Accessory Friction
309
Table 10.2 Coefficients for Valve Train Friction Terms
Flat
follower
𝑐f f
(kPa-mm)
Roller
follower
𝑐rf
(kPa-mmmin/rev)
Oscillating
hydrodynamic
𝑐oh
(kPa-mmmin/rev)1∕2
Oscillating
mixed
𝑐om
(kPa)
Type I
200
0.0076
0.5
10.7
Type I
133
0.0050
0.5
10.7
Type II
600
0.0227
0.2
42.8
Type III
400
0.0151
0.5
21.4
Type IV
400
0.0151
0.5
32.1
Configuration
Single overhead
cam (SOHC)
Double
overhead cam
(DOHC)
Single overhead
cam (SOHC)
Single overhead
cam (SOHC)
Cam in block
(CIB)
Type
Source: Patton et al., 1989.
Figure 10.24 Engine oil pump.
(Courtesy Melling Engine Parts.)
Figure 10.25 Engine water
pump. (Courtesy Airtex
Products.)
310
Friction and Lubrication
Table 10.3 Coefficients for Auxiliary Friction Terms
Oil pump
Water pump
𝑐1
(kPa)
𝑐2
(kPa-min/rev)
𝑐3
(kPa-min2 /rev2 )
n
1.28
0.13
0.0079
0.002
−8.4 × 10−7
+3.0 × 10−7
0.3
0.7
Source: Shayler et al., 2005.
The accessory mean effective pressure with a viscosity correction suggested by Shayler
(2005) is
amep = 𝑐1 + (𝑐2 𝑁 + 𝑐3 𝑁 2 )(𝜇∕𝜇ref )𝑛
(10.46)
where the coefficients are given in Table 10.3.
10.10 PUMPING MEAN EFFECTIVE PRESSURE
The pumping mean effective pressure (pmep) is the sum of the pressure drops across flow
restrictions during the intake and exhaust strokes. It is a measure of the work required to
move the fuel--air mixture into and out of an engine. The flow restrictions are categorized
into four main areas: the intake system (is), the inlet valves (iv), the exhaust valves (ev), and
the exhaust system (es). The major intake system flow restrictions are the throttle valve,
air filter, intake manifold, and carburetor. The exhaust system flow restrictions include the
exhaust manifold, catalytic converter, muffler, and tail pipe. Accordingly, the total pmep
is given by Equation 10.47:
pmep = Δ𝑃is + Δ𝑃iv + Δ𝑃ev + Δ𝑃es
(10.47)
The pressure drop in the intake system Δ𝑃is
(10.48)
Δ𝑃is = 𝑃a − 𝑃𝑖
where 𝑃𝑖 is the manifold pressure upstream of the inlet valves and 𝑃a is the atmospheric
pressure. The pressure drop across the inlet valves scales with the density, mass flow rate,
and the open valve area
(
)2
1
𝑚̇
2
(10.49)
Δ𝑃iv ∼ 𝜌𝑈 ∼
𝜌 𝑛iv 𝐴iv
where 𝑛iv is the number of intake valves per cylinder. The mass flow rate scales with the
volumetric efficiency that in turn is proportional to the intake/atmospheric pressure ratio:
𝑚̇ ∼ 𝑉d 𝑁 ∼ 𝑒v 𝑏2 𝑠𝑁 ∼
𝑃𝑖 2
𝑏 𝑈̄ p
𝑃a
(10.50)
Neglecting the density change across the valves, the inlet valve pressure drop therefore
scales as the square of the mean piston speed
(
)2
2
𝑃𝑖 𝑈̄ p 𝑏
(10.51)
Δ𝑃iv = 𝑐v
𝑃a 𝑛iv 𝐷2
iv
Overall Engine Friction Mean Effective Pressure
311
where 𝐷iv is the intake valve diameter. Similarily, the exhaust valve pressure drop Δ𝑃ev
scaling is
(
)2
2
𝑃𝑖 𝑈̄ p 𝑏
(10.52)
Δ𝑃ev = 𝑐v
2
𝑃a 𝑛ev 𝐷ev
where 𝐷ev is the exhaust valve diameter, and 𝑛ev is the number of exhaust valves per
cylinder. The constant of proportionality determined by Millington and Hartles (1968) for
small high-speed diesel engines is 𝑐v = 4.12 × 10−3 kPa-s2 /m2 .
The exhaust system pressure drop Δ𝑃es also scales with the square of the mass flow
rate:
(
)2
𝑃𝑖
𝑚̇ 2
̄
Δ𝑃es ∼
= 𝑐es
(10.53)
𝑈
𝜌es
𝑃a p
If the total exhaust system pressure drop is assumed to be 40 kPa at a piston speed
of 15 m/s at wide open throttle conditions, then the proportionality constant 𝑐es equals
0.178 kPa-s2 /m2 (Patton et al., 1989).
10.11 OVERALL ENGINE FRICTION MEAN EFFECTIVE PRESSURE
The preceding component analyses can be combined to form an overall engine friction
mep model. The component equations have been used to develop a fmep program,
Friction.m that is included in Appendix F.16. The effect of throttling on fmep is
examined in a chapter homework problem.
EXAMPLE 10.1
Friction Mean Effective Pressure
What are the crankshaft, piston, valve train, pumping, and accessory fmep for the fourcylinder engine specified in Table 10.4, if it is operated at wide open throttle (WOT) at
speeds varying from 1000 to 6000 rpm?
SOLUTION The input parameters for a SOHC four-cylinder in-line engine with two intake and
two exhaust valves per cylinder are given in Table 10.2. Note that a wide open throttle
condition is one in which the intake pressure is the same as the atmospheric pressure. Using
the Friction.m program, the fmep components are plotted as a function of engine speed
in Figure 10.26. The friction from the piston rings and skirt is the largest component. As
the engine speed increases from 1000 to 6000 rpm, the total fmep increases nonlinearly
from about 100 to 260 kPa, with a large increase in the pumping friction proportion.
A quadratic correlation of the total fmep for this engine is given by Equation 10.54:
(
fmep = 94.8 + 2.3
)
(
)
𝑁 2
𝑁
+ 4.0
1000
1000
(10.54)
The relative contributions of the crankshaft, piston, valve train, accessory, and pumping
to the overall fmep are plotted in Figure 10.27. For wide open throttle conditions at low
engine speeds, the overall fmep is primarily due to piston and valve train friction. As the
engine speed increases, the pmep fraction increases to about 35%, and the valve train fmep
fraction decreases from 35 to 10%.
312
Friction and Lubrication
Table 10.4 Representative Engine Parameters for Example 10.1
Bore (mm)
Stroke (mm)
Number of cylinders
Compression ratio
82.5
82
4
19.4
Atmospheric pressure (kPa)
Intake pressure (kPa)
Exhaust pressure (kPa)
101
101
103
Intake valves/cylinder
Exhaust valves/cylinder
Intake valve diameter (mm)
Exhaust valve diameter (mm)
Number of crankshaft bearings
Crankshaft bearing dia. (mm)
Crankshaft bearing length (mm)
1
1
36.5
31.5
5
54
21.6
Number of connecting rod bearings
Connecting rod bearing dia. (mm)
Connecting rod bearing length (mm)
Number of camshaft bearings
4
49
21.4
5
300
250
fmep (kPa)
200
ing
Pump
150
ory
Access
in
Valve tra
100
Piston
50
Figure 10.26 Friction
mean effective pressure
versus engine speed.
0
1000
Crankshaft
2000
3000
4000
Engine speed (rpm)
5000
6000
10.12 LUBRICATION
Oil is used as a lubricant to reduce the friction between the principal moving parts of an
engine. In addition to lubricating, engine oil is expected to meet a number of other service
requirements: to act as a coolant for the pistons, rings, and bearings, to enhance the rings
combustion seal, to control engine wear or corrosion, and to remove impurities from lubricated regions. To meet these requirements, additives are used with petroleum or synthetic
base oil stocks. The additives include antifoam agents, antirust agents, antiwear agents,
corrosion inhibitors, detergents, dispersants, extreme pressure agents, friction reducers,
Lubrication
313
1.00
Pumping
0.80
Fraction
Access
ory
0.60
Valve train
0.40
Piston
0.20
Figure 10.27 Relative
contributions of fmep
components versus
engine speed.
0.00
1000
Crankshaft
2000
4000
3000
Engine speed (rpm)
5000
6000
oxidation inhibitors, pour point depressants, and viscosity index improvers. Additives
range in concentration from several parts per million up to 10%.
The portion of the crude oil refiners use to make lubricants is on the order of 1% and
comes from the higher-boiling fraction and undistilled residues that possess the necessary
viscosity. Refiners use chemical processing and additives to produce oils with desirable
characteristics. Straight-run base stock from petroleum crude oil is referred to as a petroleum
oil, whereas those base stocks produced by chemical processing are called synthetic oils.
Some synthetic base stocks are compatible with petroleum base stocks and the two types
may be blended, in which case the stock is referred to as a blend.
The viscosity of a lubricating oil decreases with increasing temperature and increases
with pressure. A Newtonian oil is one in which the viscosity is independent of the shear
rate. Shear rates in engines are sometimes high enough that the viscosity decreases, and
some oils are deliberately made non-Newtonian via the introduction of polymeric materials
into low-viscosity oils. At some times during hydrodynamic lubrication, the loads increase
the oil pressure, which increases the viscosity, increasing the load capacity. It has been
suggested that this stabilizing effect is a part of the reason for effects attributed to the
‘‘property’’ oiliness. These polymeric materials also thicken oil more at high temperatures
than at low temperatures. The invention of viscosity modifiers eliminated the need to use
different viscosity oils in summer and winter operation.
The SAE classifies oils by their viscosity. Two series of grades are defined in
Table 10.5. Grades with the letter W(Winter) are based on a maximum low-temperature
dynamic viscosity, a maximum borderline pumping temperature, and a minimum kinematic
viscosity at 100◦ C. Grades without the letter W are based on a minimum and a maximum
kinematic viscosity at 100◦ C. Increasing SAE grade numbers correspond to increasing
viscosity. In terms of carbon content of the SAE grades, Gruse (1967) offers the guidelines listed in Table 10.6. Note that the SAE grade and thus the viscosity, increases with
increasing molecular weight.
The borderline pumping temperature is measured via a standard test procedure, ASTM
D3829, and is a measure of an oil’s ability to flow to an engine oil pump inlet and provide
adequate oil pressure during warm-up. This is to ensure that the oils meeting this standard
will all flow readily on cold start-up and reach moving parts as quickly as possible. The
dynamic viscosity is measured using the standard procedure SAE J300, and the kinematic
Friction and Lubrication
Table 10.5 SAE Specifications for Engine Oils
SAE
viscosity
grade
Dynamic viscosity 𝜇 (cP)
at temperature (◦ C),
maximum
Borderline pumping
temperature (◦ C),
maximum
3250 at −30
3500 at −25
3500 at −20
3500 at −15
4500 at −10
6000 at −5
-----
−35
−30
−25
−20
−15
−10
-----
0W
5W
10W
15W
20W
25W
20
30
40
50
Kinematic viscosity
𝜈 (cSt) at 100 ◦ C
Minimum
Maximum
3.8
3.8
4.1
5.6
5.6
9.3
5.6
9.3
12.5
16.3
<9.3
<12.5
<16.3
<21.9
Source: SAE Handbook, 2003.
Table 10.6 SAE Engine Oil Carbon Content
SAE 10
SAE 30
SAE 50
Range
Average
C25 --C35
C30 --C80
C40 --C100
C28
C38
C41
Source: Gruse, 1967.
viscosity is determined following ASTM D445, a procedure that measures the time required
for a given volume of oil to flow through a capillary tube. This is to ensure that oils meeting
this standard will all exhibit proper lubricating properties under the high pressures and
temperatures found at normal engine operating conditions.
As shown in Figure 10.28, a multiviscosity grade of oil is one that satisfies one of each
of the two grades at different temperatures. When the engine is cold, a relatively low oil
viscosity, such as 10W, is required, and when the engine is warmed up, there is a need for
a higher viscosity, such as 30W, for adequate lubrication and sealing. Multigrade oils have
been created to meet both cold start and normal operation requirements by adding polymeric
high molecular weight compounds that reduce the viscosity change with temperature. As
30W
20W
10W
SA
E1
Viscosity
314
Figure 10.28 Engine oil
viscosity grades.
–40
SAE 30
0W
-30
SAE 10W
SAE 50
SAE 40
SAE 30
SAE 20
SAE 10
0
40
100
Temperature (°C)
160
References
315
Table 10.7 Constants for Equation 10.55
SAE
grade
10
20
30
40
50
60
𝐶1
(N s/m2 )
𝐶2
(◦ C)
1.09 × 10−4
9.38 × 10−5
9.73 × 10−5
8.35 × 10−5
1.17 × 10−4
1.29 × 10−4
1157.5
1271.6
1360.0
1474.4
1509.6
1564.0
Source: Hamrock et al. (1999).
Figure 10.28 indicates, at low-temperatures, the viscosity of an SAE 10W30 oil will be less
than that of a SAE 30 oil to meet cold start requirements.
The dynamic viscosity 𝜇 (N s/m2 ) of SAE oils as a function of temperature is correlated
by Equation 10.55:
[
]
𝐶2
(10.55)
𝜇 = 𝐶1 exp
1.8 𝑇 (◦ C) + 127
Values of the constants 𝐶1 and 𝐶2 of Equation 10.55 for various SAE grades of engine oils
are given in Table 10.7, as a function of temperature 𝑇 (◦ C).
Finally, it should be mentioned that many two-stroke engines, especially small ones,
achieve upper cylinder lubrication by mixing oil with the gasoline. The oil used is typically
SAE 30 or 40, and formulated specifically to minimize combustion chamber deposits. In
this situation, additional control of the oil (and the fuel) is required to prevent spark fouling,
to assure miscibility with the fuel, and to provide for a hydrodynamic film of the proper
viscosity. The fuel--oil mixture that contacts the cold walls separates during compression
and combustion, leaving an oil film on the wall.
10.13 REFERENCES
ALLEN, D. G., B. R. DUDLEY, J. MIDDLETOWN, and D. A. PANKA (1976), ‘‘Prediction of Piston RingCylinder Bore Oil Film Thickness in Two Particular Engines and Correlation with Experimental
Evidences,’’ Conference on Piston Ring Scuffing, Mechanical Engineering Pub. Ltd., London,
p. 107.
ARCOUMANIS, C., M. DUSZYNSKI, H. LINDENKAMP, and H. PRESTON (1998), ‘‘Measurements of Lubricant
Film Thickness in the Cylinder of a Firing Diesel Engine Using LIF,’’ SAE paper 982435.
BISHOP, I. N. (1964), ‘‘Effect of Design Variables on Friction and Economy,’’ SAE paper 640807.
BROWN, W. L. (1973), ‘‘The Caterpiller Imep Meter and Engine Friction,’’ SAE paper 730150.
GISH, R., S. McCULLOUGH, J. RETZLOFF, and H. MUELLER (1957), ‘‘Determination of True Engine
Friction,’’ SAE paper 117.
GRUSE, W. A. (1967), Motor Oils: Performance and Evaluation, Van Nostrand Reinhold, New York.
HAMROCK, W., B. JACOBSON, and S. SCHMID (1999), Fundamentals of Machine Elements, McGraw-Hill
Publishing, New York.
LEARY, W. A. and J. U. JOVELLANOS (1944), ‘‘A Study of Piston and Piston--Ring Friction,’’ NACA
ARR-4J06.
McGEEHAN, J. A. (1978), ‘‘A Literature Review of the Effects of Piston and Ring Friction and
Lubricating Oil Viscosity on Fuel Economy,’’ SAE paper 780673.
MERRION, D. (1994), ‘‘Diesel Engine Design for the 1990’s,’’ SAE SP-1011.
MILLINGTON, B. and E. HARTLES (1968), ‘‘Frictional Losses in Diesel Engines,’’ SAE paper 680590.
316
Friction and Lubrication
NOORMAN, M., D. ASSANIS, D. PATTERSON, S. TUNG, and S. TSEREGOUNIS (2000), ‘‘Overview of Techniques for Measuring Friction Using Bench Tests and Fired Engines,’’ SAE paper 2000-01-1780.
PATTON, K. J., R. G. NITSCHKE, and J. B. HEYWOOD (1989), ‘‘Development and Evaluation of a Friction
Model for Spark Ignition Engines,’’ SAE paper 890836.
ROSENBERG, R. C. (1982), ‘‘General Friction Considerations for Engine Design,’’ SAE paper 821576.
SAE Handbook (2003), Fuels and Lubricants, Society of Automotive Engineers, Warrendale,
Pennsylvania.
SANDOVAL, D. and J. HEYWOOD (2003), ‘‘An Improved Friction Model for Spark Ignition Engines,’’
SAE paper 2003-01-0725.
SHAYLER, P., D. LEONG, and M. MURPHY (2005), ‘‘Contributions to Engine Friction During Cold, Low
Speed Running and the Dependence on Oil Viscosity,’’ SAE paper 2005-01-1654.
TAYLOR C. (1985), The Internal Combustion Engine in Theory and Practice, Vol. 2, MIT Press,
Cambridge, Massachusetts.
TING, L. L. and J. E. MAYER, Jr. (1974), ‘‘Piston Ring Lubrication and Cylinder Bore Analysis,
Part I Theory and Part II Theory Verification,’’ J. Lubr. Tech., Vol. 96, pp. 305--314.
TSEREGOUNIS, S., M. VIOLA, and R. PARANJPE (1998), ‘‘Determination of Bearing Oil Film Thickness (BOFT) for Various Engine Oils in an Automotive Gasoline Engine Using Capacitance
Measurements and Analytical Predictions,’’ SAE paper 982661.
10.14 HOMEWORK
10.1
How is the fmep computed from the data in Figure 10.15?
10.2
It has been observed that the oil film thickness on the cylinder wall tends to be greater at
lower loads. Relate this observation to the fmep versus load curves in Figure 10.6.
10.3
A four-cylinder engine has a 85 mm bore 𝑏 and 90 mm stroke 𝑠, and operates at 5000 rpm.
What is the power lost to piston skirt friction?
10.4
What is the power (𝑊 ) lost to main crankshaft friction in a four-cylinder engine with a
100 mm bore and stroke if the engine is operating at 1500 rpm? Assume the main crankshaft
has five bearings, each with a diameter of 50 mm and length of 25 mm.
10.5
Racing mechanics will often modify an engine by increasing the bearing clearances above
the manufacturer specifications. This can increase the power of an engine. With reference
to Equation 10.2, discuss why the power is increased, and the tradeoffs that should be
considered in choosing a clearance.
10.6
Using Petrov’s equation, Equation 10.22, show that the nondimensional fmep of journal bearings is inversely proportional to the oil Reynolds number, as indicated by
Equation 10.24.
10.7
Using a piston force balance, discuss the zero crossings in Figure 10.14, and the absence
of zero crossings between 0 and 180◦ .
10.8
With reference to a piston force balance, how is the piston and ring friction affected by the
mass of the piston?
10.9
For the piston specifications given in Table 10.4, at what engine speed will the piston skirt
friction be equal to the piston ring friction?
10.10
What are the pumping, accessory, valve train, piston, and crankshaft fmeps for a sixcylinder engine at 3000 rpm and WOT with 0.1-m bore and stroke, and compression ratio
of 11? The number of crankshaft bearings is seven. Assume the other engine specifications
are as given in Table 10.4.
Homework
317
10.11
Compute the fmep components (pumping, accessory, valve train, piston, and crankshaft)
for the engine specified in Table 10.4 but throttled to 50 kPa inlet pressure. Plot the results
versus engine speed from 1000 to 6000 rpm.
10.12
If one cylinder of a multicylinder spark ignition engine is motored by disconnecting the
spark plug, it has been observed that the motored cylinder pressure increases with load.
Discuss why this is to be expected.
10.13
Oil companies are advertising ‘‘slippery oils,’’ that is, non-Newtonian, that if used in your
automobile will slightly reduce the fuel consumption. The implication is that when the
slippery oil is compared with conventional oil of equal viscosity, the slippery oil will have
a reduced friction coefficient. How is this possible?
Chapter
11
Heat and Mass Transfer
11.1 INTRODUCTION
Satisfactory engine heat transfer is required for a number of important reasons, including
material temperature limits, lubricant performance limits, emissions, and knock. Since the
combustion process in an internal combustion engine is not continuous, as is the case for an
external combustion engine, the time average component temperatures are much less than
the peak combustion temperatures. However, the temperatures of certain critical areas need
to be kept below material design limits. Aluminum alloys begin to melt at temperatures
greater than 775 K, and the melting point of iron is about 1800 K. Temperature gradients
around the cylinder bore will cause bore distortion and subsequent increased blowby, oil
consumption, and piston wear. Cooling of the engine cylinder is also required to prevent
knock from occuring in spark ignition engines.
Exhaust system heat transfer is also an important factor in emissions and exhaust turbine
performance. Satisfactory catalytic converter performance only occurs above a threshold or
light-off temperature. The threshold temperature (catalyst oxidation efficiency greater than
50%) for the catalyzed oxidation of hydrocarbon and carbon monoxide emissions is about
500 K so that at exhaust temperatures less than 500 K, catalytic converter performance is
adversely affected. In addition, the continued oxidation of hydrocarbons and other pollutants
in the exhaust system is a function of the exhaust system temperature.
Heat transfer to the airflow in the intake manifold is also an important consideration.
The heat transfer lowers the volumetric efficiency, since the density of the intake air is
decreased. Plastic intake manifolds with reduced thermal conductivity (as well as reduced
weight) are now being used to reduce intake air heating.
The heat transfer rate in an engine is dependent on the coolant temperature and the
engine size, among other variables. There are complex interactions between the various
engine parameters. For example, as the temperature of the engine coolant decreases, the heat
transfer to the coolant will increase, and the combustion temperature will decrease. This will
cause a decrease in the combustion efficiency and an increase in the volumetric efficiency.
It will also cause an increase in the thermal stresses in the cylinder sleeve, and increase the
size of the radiator needed, since the coolant ambient temperature difference will decrease.
The formation of nitrogen oxides will decrease, and the oxidation of hydrocarbons will
decrease. The exhaust temperature will also decrease, causing a decrease in the performance
of the catalytic converter and a turbocharger.
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
318
Engine Cooling Systems
319
11.2 ENGINE COOLING SYSTEMS
There are two types of engine cooling systems used for heat transfer from the engine
block and head liquid cooling and air cooling. With a liquid coolant, thermal energy is
removed through the use of internal cooling channels within the engine block, as shown
schematically in Figure 11.1. With air as a coolant, thermal energy is removed through the
use of fins attached to the cylinder wall, as shown schematically in Figure 11.2. Both types
of cooling systems have various advantages and disadvantages. Liquid systems are much
quieter than air systems, since the cooling channels absorb the noise from the combustion
processes. However, liquid systems are subject to freezing, corrosion, and leakage problems
that do not exist in air systems.
As indicated in Figure 11.1, the water cooling system is usually a single loop where a
water pump sends coolant to the engine block and then to the head. The coolant will then
flow to the top of a radiator or heat exchanger and exit at the bottom to flow back to the
pump. Early engines, such as the Ford Model T engine, did not have a water pump, and
used a thermosyphon natural convection loop to circulate the coolant from the engine to
the radiator. The water pump is located at the bottom of the engine to minimize cavitation.
The pump is usually powered by the engine; however, electric water pumps have also been
used. During engine warm-up, a thermostatically controlled valve will recycle the coolant
flow through the engine block, bypassing the heat exchanger. As the engine heats up, the
valve will open up, and allow the coolant to flow to the radiator. The time required for
engine warm-up to a steady-state operating temperature depends on the engine size, speed,
and load and is typically of the order of 10 min for an automotive engine. Dual circuit
cooling with separate circuits to the head and block has also been used.
The design of the liquid cooling passages in the engine block and head is done empirically. The primary design consideration is to provide for sufficient coolant flow at the
high heat flux regions, such as the exhaust valves. Since the area between exhaust valves is
difficult to cool, some automotive engine designs use only one exhaust valve to reduce the
heating of the inlet air--fuel mixture, and thus increase the volumetric efficiency. A review
of precision cooling considerations is given in Robinson et al. (1999).
Head
Radiator
Engine Cylinder
heat
block
Figure 11.1 Liquid cooling system
schematic.
Heat
rejected
Water
pump
Air
coolant
Cylinder
heat
Q
A
Figure 11.2
Air cooling system schematic.
320
Heat and Mass Transfer
The boiling point temperature of the liquid coolant can be raised by increasing the
pressure or by adding an additive with a high boiling point, such as ethylene glycol.
The heat fluxes and surface temperatures near the exhaust manifold and port are high
enough so that nucleate boiling can occur in the coolant at those locations. The boiling heat
transfer coefficients are much larger than single-phase forced convection, so that the surface
temperatures will be correspondingly lower. For heat fluxes of the order of 1.5 MW/m2 ,
the resulting surface temperature of the cooling jacket will be about 20--30◦ C above the
saturation temperature, which is typically 130◦ C (400 K). The nucleate boiling process is
very complex, as bubbles formed on the cooling channel surface are swept downstream
and then condensed in cooler fluid. Engines with relatively low-power output, less than
20 kW, primarily use air cooling, shown schematically in Figure 11.2. Because the thermal
conductivity of air is much less than that of water, air systems use fins to lower the air-side
surface temperature. For higher power output, an external cooling fan is used to increase the
air-side heat transfer coefficient. Aircraft engines are, for the most part, air cooled; supplying
the required airflow is not a problem, since the engine need not be enclosed and is usually
located right behind a propeller. Engines that are operated for very short periods of time,
such as engines used in 1/4-mile dragsters, do not use a cooling system but use the thermal
capacitance of the engine block to keep the gas-side surface temperatures within limits.
Since about one-third of the fuel energy is lost as heat transfer to the coolant, it
would seem reasonable to try to reduce this heat loss, thereby increasing the efficiency
of the engine. One way to reduce the heat flow to the coolant is to increase the thermal
resistance of the engine block through the use of lower thermal conductivity materials,
such as ceramics, or adding thermal insulation to the engine. Ceramic wall materials that
can operate at higher temperatures and have a lower thermal conductivity than cast iron are
silicon nitride and zirconia.
Experimental results from such engines show that a reduction in the coolant heat loss
does not result in a corresponding increase in the efficiency of the engine (Sun et al.,
1993). There are a number of reasons for this. First, by insulating the engine, the average
cylinder temperature increases, and the exhaust temperature and enthalpy increase. The
thermal energy that had been conducted to the coolant is now added to the exhaust stream.
Secondly, since the penetration depth of the combustion heat flux is only about a millimeter,
the coolant heat transfer is a relatively steady-state process throughout the cycle, but the
positive work is produced only during the expansion stroke. The coolant heat transfer
that occurs during the other strokes is not available to be converted to work. Thirdly, the
higher wall temperatures will heat the incoming gas during the intake stroke lowering its
volumetric efficiency. For spark ignition engines, the higher wall temperatures during the
compression stroke can give rise to knock problems. Therefore, the majority of the engines
that have employed increased cylinder thermal resistance are compression ignition engines.
11.3 ENGINE ENERGY BALANCE
An engine energy balance is obtained through experiments performed on instrumented engines. Figure 11.3 depicts an engine instrumented to determine the quantities of heat rejected
to oil, water, and to the ambient air. Flow meters are installed in the water, and oil circuits
and thermocouples measure the inlet and outlet temperatures. An energy balance, Equations
11.1 and 11.2, applied to the water coolant and the oil flowing through the engine yields
̇ p )water (𝑇3 − 𝑇4 )
(11.1)
𝑄̇ water = (𝑚𝑐
̇ p )oil (𝑇1 − 𝑇2 )
𝑄̇ oil = (𝑚𝑐
(11.2)
Engine Energy Balance
m fuel
mex
Qamb
mair
321
7
5
Insulated
exhaust
plenum
8
6
Water flow
meter
Oil flow
meter
2
3
mwater
4
Water
cooler
moil
Water
pump
Oil
pump
Oil
cooler
in
Lab
water
out
1
Wshaft
in
Lab
water out
Figure 11.3 Engine instrumented for energy balance measurements.
Determining the heat loss to the ambient air is more involved. The first law, Equation
11.3, applied to the engine system is
̇ air + (𝑚ℎ)
̇ fuel − (𝑚ℎ)
̇ exh
𝑄̇ amb = (𝑚ℎ)
−𝑄̇ water − 𝑄̇ oil − 𝑊̇ shaf t
(11.3)
The mass flow rate of the exhaust is known in terms of the measured mass flow rates
of air and fuel since
𝑚̇ exh = 𝑚̇ air + 𝑚̇ fuel
(11.4)
The enthalpies of the exhaust, the air, and the fuel are based on the measured temperatures 𝑇5 , 𝑇6 , and 𝑇7 , respectively. The exhaust composition can be calculated theoretically
from the known fuel--air equivalence ratio or it may be measured. In either case, it is
important that the temperature 𝑇5 corresponds to the mass-averaged temperature of the
exhaust; for this reason, there is an insulated plenum that serves to mix the hot exhaust gas
emitted early in the cycle with the cool exhaust gas emitted later in the cycle. A further
complication is that the thermocouple measurement must be corrected for radiation heat
transfer to obtain the true gas temperature. An energy balance on the thermocouple tip,
Equation 11.5, yields
𝑇exh = 𝑇5 +
𝜖𝜎 4
(𝑇 − 𝑇84 )
𝒉 5
(11.5)
where 𝜖 is the emissivity of the thermocouple tip, 𝜎 is the Stefan--Boltzmann constant, h is
the heat transfer coefficient at the tip (printed in bold to distinguish it from the symbol ℎ for
enthalpy), 𝑇5 is the tip temperature, and 𝑇8 is the exhaust plenum inner wall temperature.
In doing these energy balances, it is common practice to evaluate the maximum heat
that can be recovered from the exhaust gas. This is computed from an energy balance,
322
Heat and Mass Transfer
Table 11.1 Energy Balance on a Medium Speed, Four-Stroke, Turbocharged Diesel Engine (All
energy rates are normalized by the fuel rate, 𝑄̇ in )
𝑁
(rpm)
bmep
(bar)
𝑄̇ exh
𝑄̇ water
𝑄̇ oil
𝑄̇ amb
𝑊̇ shaf t
𝑊̇ friction
𝑄̇ loss
500
500
400
9.90
3.52
3.50
0.459
0.437
0.432
0.118
0.108
0.151
0.037
0.065
0.074
0.069
0.092
0.026
0.317
0.298
0.315
0.100
0.178
0.092
0.124
0.087
0.159
Source: Whitehouse(1970).
Equation 11.6, on the exhaust where it is cooled to ambient temperature:
𝑄̇ exh = 𝑚̇ exh [ℎexh (𝑇exh ) − ℎexh (𝑇amb )]
(11.6)
In evaluating the exhaust enthalpy at ambient temperature, equilibrium water quality should
be used, as discussed in Chapter 4. If Equation 11.6 is substituted into Equation 11.3, one
obtains
𝑄̇ amb = 𝑄̇ in − 𝑄̇ exh − 𝑄̇ water − 𝑄̇ oil − 𝑊̇ shaf t
(11.7)
where, by definition
𝑄̇ in = (𝑚ℎ)
̇ air + (𝑚ℎ)
̇ fuel − (𝑚ℎ)
̇ exh
(11.8)
Finally, if the fuel and air are at an ambient temperature, the engine runs lean or
stoichiometric, and the ambient temperature and pressure are coincident with the reference
temperature and pressure, then the inlet enthalpy is the product of the fuel flow rate and the
fuel’s stoichiometric heat of combustion.
Some results obtained by Whitehouse (1970) for a medium-speed diesel engine are
given in Table 11.1. The terms in the table are normalized by the input energy of the
fuel 𝑄̇ in . The diesel engine used was a single-cylinder test engine with a bore of 304.8
mm, a stroke of 381 mm, and a compression ratio of 12.85. The table also gives the heat
equivalence of the friction so that one can ascertain how much of the heat lost to the ambient
air, to the oil, and to the water, is from the working fluid and how much is from friction.
The overall heat loss is the sum of the heat transfer to the water, oil, and ambient air minus
the friction work:
𝑄̇ loss = 𝑄̇ water + 𝑄̇ oil + 𝑄̇ ambient − 𝑊̇ friction
(11.9)
Inspection of the shaft work term reveals that the engine has a brake thermal efficiency
of about 30%. About 45% of the energy is rejected in the exhaust, 10--15% is dissipated by
friction, and 10--15% is dissipated with heat loss.
We now examine representative energy balance results for spark ignition engines.
Figure 11.4 shows the results of an energy balance on a small, spark ignition automobile
engine. This engine has an internal oil pump, and the heat rejected to the oil is carried
away partly by the coolant and partly by the heat lost to ambient air. As the load increases
and the intake manifold pressure increases from 𝑃i = 0.4 bar to 𝑃i = 0.8 bar, the energy
converted to shaft work increases from about 20 to 30%, the coolant load decreases from
about 40 to 30%, the exhaust energy varies from about 30 to 35%, and the heat lost to
ambient air decreases from about 10 to 5%. The energy dissipated by friction decreases
from about 14 to 7% for the same loads. The total heat loss from the gas to the coolant and
ambient air during the cycle is about 28--36%.
323
Engine Energy Balance
Energy balance at 3000 rpm
110.0
100.0
Heat loss
to ambient
Percentage of liberated fuel energy
90.0
80.0
Exhaust
energy
70.0
60.0
50.0
Coolant
load
40.0
Friction dissipation
30.0
20.0
Shaft
work
10.0
Figure 11.4 Energy balance
on an automotive engine.
(Courtesy D. Brigham of Ford
Motor Co.)
0.0
0.4
0.5
0.6
0.7
0.8
0.9
Intake manifold pressure (bar)
Ryder (1950) performed an energy balance on a Pratt and Whitney Mark R-2800 aircooled aircraft engine, as shown in Table 11.2. Under cruise conditions, the engine runs
at 1800 rpm, with the engine fueled slightly lean at 𝜙 = 0.90, producing a bmep = 8.75
bar, and brake thermal efficiency of 0.98 × 29% = 28%. During takeoff, the engine speed
is increased to 2700 rpm, and the engine is fueled extremely rich with 𝜙 = 1.65. Only
about 46% of the fuel’s heat of combustion is released, producing a bmep = 13.72 bar. This
is done to utilize the liquid fuel’s latent heat for cooling and to avoid knock limiting the
power. Because the fuel consumed during takeoff is small compared with that used in the
entire trip, the fact that fuel is wasted is of secondary concern. Notice that during takeoff,
35% of the heat released is converted to shaft work, and the brake thermal efficiency of the
engine is reduced to 0.46 × 35% = 16.1%.
Upon comparison, we see that the diesel engine loses only about one-half as much
heat to the coolant and ambient air as the gasoline engine, yet their shaft efficiencies are
Table 11.2 Energy Balance on an Air-Cooled, Spark Ignition Aircraft Engine
𝜙
N
(rpm)
bmep
(bar)
𝑄̇ exhaust
𝑄̇ oil
𝑄̇ ambient
𝑊̇ shaf t
𝑄̇ in ∕𝑚̇ f 𝑞c
0.90
1.65
1800
2700
8.75
13.72
0.44
0.44
0.09
0.08
0.18
0.12
0.29
0.35
0.98
0.46
Source: Ryder (1950).
All energy rates are normalized by the fuel rate, 𝑄̇ in .
Engine: 𝑏 =146.1 mm, 𝑠 = 152.4 mm.
324
Heat and Mass Transfer
about equal since the diesel engine has more exhaust heat loss than the gasoline engine. As
discussed above, experiments with insulated engines show that a reduction in the coolant
heat loss has a small impact on the shaft efficiency and that the energy no longer lost to the
coolant mostly appears in the exhaust flow. If a turbocharger is used, the available portion
of the exhaust energy can be converted to useful work.
11.4 CYLINDER HEAT TRANSFER
There are a wide range of temperatures and heat fluxes throughout an internal combustion
engine. The value of the local heat flux can vary by an order of magnitude depending on
the spatial location in the combustion chamber and the relative crank angle. The sources
of the heat flux are not only the hot combustion gases, but also the friction that occurs
between moving surfaces, such as the piston rings and the cylinder wall. When an engine is
running at a steady state, the heat transfer rates throughout most of the engine structure are
relatively constant. As will be shown, unsteady periodic effects are limited to a penetration
layer about 1--5 mm thick at the combustion gas cylinder wall interface.
Experiments indicate that the heat flux increases with increasing engine load and speed,
with the maximum heat flux through the engine components occurring at fully open throttle
and at maximum speed. Peak heat fluxes are on the order of 1--10 MW/m2 . The heat flux
is largest in the center of the cylinder head, the exhaust valve seat, and the center of the
piston. About 50% of the heat flow to the engine coolant is through the engine head and
valve seats, 30% through the cylinder sleeve or walls, and the remaining 20% through the
exhaust port area.
The piston and valves, since they are moving, are difficult to cool, and operate at the
highest temperatures. Temperature measurements indicate that the greatest temperatures
occur at the top, that is, the crown of the piston, since it is in direct contact with the
combustion gases. The crown temperatures can be as high as 550 K. The temperatures of
the piston and valves depend on their thermal conductivity. As the thermal conductivity
increases, the conduction resistance decreases, resulting in lower surface temperatures. For
the same speed and loading, aluminum pistons are about 40 K cooler than cast iron pistons.
The main cooling paths for the piston are conduction through the piston rings to the
cylinder wall and conduction through the piston body to the air--oil mist on the underside of
the piston, as shown in Figure 11.5. About half of the heat rejected to the cylinder wall from
the piston is from cylinder friction. The main cooling path for the exhaust valves is through
the valve seat, since the exhaust valve is closed for about three strokes of the four-stroke
cycle. The cooling mechanism is thermal contact conduction. The conductance depends on
the maximum cylinder pressure, which compresses the valve onto the valve seat. Values
of the thermal contact conductance between 5000 and 35,000 W/mK have been measured
(Wisniewski, 1998). Hollow valve stems partially filled with sodium have been used to
increase the effective axial thermal conductivity of the exhaust valve. The sodium melts
at 370 K, so at temperatures greater than 370 K, there is internal natural conduction heat
transfer in the axial direction inside the valve stem.
In order to accurately calculate the total heat loss from the combustion chamber, one
must evaluate the heat flux at every location in the combustion chamber and integrate
over the active cylinder area, as indicated by Equation 11.10. For this reason, a number of
surface thermocouples are installed into the engine at representative locations.
̇ =
𝑄(𝑡)
∫A(t)
𝑞 ′′ (𝑡)𝑑𝐴
(11.10)
Cylinder Heat Transfer
325
Qgas
Qrings
Coolant
Piston
Qskirt
Oil mist
Figure 11.5
Piston cooling paths.
2
Figure 11.6 Cylinder
head heat flux profiles
of five consecutive
combustion cycles in a
spark ignition engine.
Adapted from Alkidas
and Myers (1982).
Surface heat flux, MW/m2
Cycle
1
2
3
4
5
1.5
1
0.5
0
–0.5
0
100
200
300
400
500
Crank angle (deg)
600
700
800
Resolution of the instantaneous heat transfer at the cylinder surface can be achieved
by inserting a surface thermocouple into the engine structure. Figure 11.6 shows the heat
flux measurements made by Alkidas and Myers (1982) using a surface thermocouple in a
cylinder head of a propane-fueled engine. Five consecutive combustion cycles are shown,
with a peak heat flux of about 1.8 MW/m2 . Note that most of the heat transfer occurs early in
the expansion stroke from 360◦ to 420◦ crank angle, when the combustion gas temperatures
are greatest. Negative heat flux, that is, from cylinder surface to combustion gases, can
occur late in the expansion stroke and during the intake stroke. The cycle-to-cycle variations
noted in Figure 11.6 are caused by cycle-to-cycle variations in arrival times of the turbulent
flame at the thermocouple.
The surface thermocouple used for instantaneous heat flux measurements was originally developed by Bendersky (1953). The essential features of a surface thermocouple are
shown in Figure 11.7. Within the plug are two iron--constantan thermocouple junctions,
one at the surface and one at a depth Δ𝑥 from the surface. The basic idea is that according
to Fourier’s law, Equation 11.11, for small Δ𝑥,
𝑞 ′′ = −𝑘
Δ𝑇
Δ𝑥
(11.11)
326
Heat and Mass Transfer
Steel rod
Iron wire
Constantan wire
Epoxy seal
Iron–constantan
thermocouple
Aluminum plug
"Pyrex" tube
Figure 11.7 Surface
thermocouple plug
used to measure
instantaneous heat flux.
Adapted from Dent
and Suliaman (1977).
"Pyrex" seal
Iron
Magnesium fluoride
Constantan
The criterion for small Δ𝑥 is that it be small compared with the thermal penetration
layer 𝛿. Unfortunately, it is just not practical to build a plug with Δ𝑥 ≪ 𝛿. Therefore,
instead, one solves the heat conduction equation between the two thermocouples, assuming
that it is one-dimensional. The measured surface temperature variation in time for one cycle
is curve-fitted by a Fourier series, shown in Equation 11.12, with Fourier coefficients 𝐴i
and 𝐵i :
𝑇 (0, 𝑡) = 𝑇̄ (0) +
𝑁
∑
[𝐴i cos(𝑖𝜔𝑡) + 𝐵i sin(𝑖𝜔𝑡)]
(11.12)
i=1
The heat flux is then given by Equation 11.13:
]
[
𝑁 √
∑
̄
Δ
𝑇
𝑖𝜔
𝑞 ′′ = −𝑘
+
[(𝐵i − 𝐴i ) sin(𝑖𝜔𝑡) + (𝐵i + 𝐴i ) cos(𝑖𝜔𝑡)]
Δ𝑥 i=1
2𝛼
(11.13)
The parameter 𝜔 is one-half the engine frequency for a four-stroke engine and equal to the
engine frequency for a two-stroke engine.
11.5 HEAT TRANSFER MODELING
The heat transfer processes in an internal combustion engine can be modeled with a variety
of methods. These methods range from simple thermal networks to multidimensional
differential equation modeling. The choice of modeling method involves considerations
such as the computational accuracy required and the data that are available to model the
engine configuration and operation.
Thermal network models, using resistors and capacitors, are very useful for rapid and
efficient estimation of the conduction, radiation, and convection heat transfer processes in
engines. Using a thermal network, the significant resistances to heat flow, and the effects
of changing material thermal conductivity, thickness, and coolant properties can be easily
determined. Thermal networks can be used directly for convection and conduction heat
Heat Transfer Modeling
327
Cylinder
wall
Tg
Figure 11.8
Tc
1
hg
Three resistor thermal network.
L
k
1
hc
transfer, and the radiation heat transfer equation needs to be linearized to conform to the
resistance model.
A simple three resistor series network, which includes convection and conduction
resistances shown in Figure 11.8, is an illustration of the steady-state heat transfer from
the engine cylinder gas to the coolant. This series path is composed of convection through
the cylinder gas boundary layer, conduction across the cylinder head wall, and convection
through the coolant liquid boundary layer. The cylinder gas boundary layer insulates the
cylinder wall from the high-temperature cylinder gases. From Fourier’s equation, Equation
11.14, the conduction resistance is
𝑅cond =
Δ𝑇
𝐿
=
𝑄∕𝐴
𝑘
(11.14)
and using Newton’s equation, Equation 11.15 the convection resistance is
𝑅conv =
Δ𝑇
1
=
𝑄∕𝐴 h
(11.15)
Examples of resistor--capacitor thermal networks applied to engine warm-up and steadystate operation are given in Shayler et al. (1993) and Bohac et al. (1996).
We now analyze the unsteady nature of the heat flux from the combustion gas to the
cylinder wall. The cylinder wall has a periodic heat flux on the gas side and a constant
surface temperature on the coolant side. The problem posed requires solution of the transient
heat conduction equation, Equation 11.16
𝜕2𝑇
𝜕𝑇
=𝛼
𝜕𝑡
𝜕𝑥2
(11.16)
subject to the following boundary conditions
−𝑘
𝜕𝑇
𝜕𝑥
′′
′′
= 𝑞0 + 𝑞1 sin(𝜔𝑡)
𝑇 = 𝑇𝐿
at 𝑥 = 0
at 𝑥 = 𝐿
(11.17)
as well as an initial condition
𝑇 = 𝑇i (𝑥) at 𝑡 = 0
(11.18)
An exact solution can be written in closed form but it is quite cumbersome and as a
result, not very illustrative. Fortunately, an approximate solution can be derived for the
practical case where
𝜔𝑡 ≫ 1 and
𝜔𝐿2
≫1
2𝛼
(11.19)
In this case, the temperature field is given by Equation 11.20:
]
(
[(
)
)
( )1∕2
𝑞0′′
𝑞1′′
𝜔 1∕2
𝜋
𝜔
𝑥 sin 𝜔𝑡 −
𝑥−
exp −
𝑇 = 𝑇𝐿 +
(𝐿 − 𝑥) +
(11.20)
𝑘
2𝛼
2𝛼
4
(𝛼∕𝜔)1∕2
328
Heat and Mass Transfer
Inspection of this solution shows that
• the surface temperature at 𝑥 = 0 oscillates with the same frequency as the imposed heat
flux but with a phase difference of 𝜋∕4;
• the amplitude of the oscillations decays exponentially with the distance 𝑥 from the
surface; the amplitude is reduced to 10% of that at the surface at a penetration distance
𝛿 given by Equation 11.21
𝛿 = −ln(0.10) (2𝛼∕𝜔)1∕2 = 2.3 (2𝛼∕𝜔)1∕2
(11.21)
For a two-stroke engine operating at 2000 rpm (𝜔 = 209 s−1 ) and made of cast iron
(𝛼 = 21 × 10−6 m2 /s), the penetration distance 𝛿 = 0.7 mm, and for aluminum, 𝛿 = 2.2 mm.
The penetration distance 𝛿 is a measure of how far into the material fluctuations about
the mean heat flux penetrates. For distances 𝑥 greater than 𝛿, the temperature profile is
more or less steady and driven only by the time-average heat flux. Since the length 𝛿 is
small compared with the dimensions (wall thickness, bore, etc.) over which conduction
heat transfer occurs, two simplifications can be made:
• Conduction heat transfer in the various parts can be assumed steady and driven by the
average flux.
• Heat transfer from the gas can be coupled to the conduction analysis accounting for
capacitance only in a penetration layer of thickness 𝛿 in series with a resistance computed
or measured for a steady state.
A five node thermal network for a cylinder wall is given in Figure 11.9. The modeling
of the penetration layer can be complicated by the presence of an oil film or deposits.
Fortunately, an accurate model is not required as the fluctuations about the mean 𝑇̄𝛿 tend
Cylinder
gas
Coolant
(air or water)
Tc, hc
Tg, hg
Ts,g
T
Penetration
layer of
thickness
Ts,g
Tg
1
hg
Figure 11.9 Thermal network with
capacitance node for penetration layer.
Tc
R
T
R
Fouling
Deposits
Rt
Tc
C
Rt = conduction path resistance
to coolant
R = /2k
C = c
Heat Transfer Modeling
329
to be small compared with the gas penetration depth temperature difference 𝑇g − 𝑇̄𝛿 . For
an engine operated at a steady state, the penetration layer is thin because the engine
frequency, which dictates the frequency components of the heat flux imposed on the gas-solid interfaces, is rather high. On the other hand, in the case of an engine being accelerated
or decelerated, the penetration layer is thicker because lower frequency components are
added to the heat flux that are characteristic of the rates of change of engine speed. For
example, one could define a characteristic time 𝜏 by
𝜏 −1 =
1 𝑑𝜔
𝜔 𝑑𝑡
(11.22)
The penetration distance 𝛿 at time 𝜏 is
𝛿 = 2.3 (2𝛼𝜏)1∕2
(11.23)
For 𝜏 = 5 s, the penetration distance 𝛿 = 33 mm for cast iron and is no longer small compared
with the typical dimensions over which the heat is transferred. Therefore, to accurately
model heat transfer in engines operated on a transient mode, the three dimensional and
unsteady features should be included.
Determination of the temperature profile of an engine component such as the piston
requires solution of the three-dimensional heat conduction equation. As mentioned earlier,
the piston can be treated as steady and driven by an average heat flux, since the penetration
layers are small. The mean cylinder gas temperature is computed using a cycle simulation to
predict instantaneous gas temperatures which are then integrated over crank angle according
to Equation 11.24:
𝑇̄g =
1
4𝜋̄𝐡g ∫0
4𝜋
𝐡g 𝑇g 𝑑𝜃
(11.24)
where 𝐡g is the instantaneous heat transfer coefficient (the determination of which is the
subject of the next section). Likewise, an average heat transfer coefficient, Equation 11.25,
is used in estimating the heat transfer coefficients on the crown of the piston in contact
with the cylinder gas.
̄𝐡g = 1
4𝜋 ∫0
4𝜋
(11.25)
𝐡g 𝑑𝜃
Results obtained by Li (1982) for the combustion gas and piston temperatures of a
2.5 L, four-cylinder engine at WOT, are given in Table 11.3 and in Figure 11.10. Notice that
both the mean combustion gas temperature and the mean heat transfer coefficient increase
with engine speed. The mean gas temperature increases because there is less time for the
Table 11.3 Variation of the Mean Gas Temperature and Heat
Transfer Coefficient at the Top of the Piston with
Engine Speed
Engine speed (rpm)
𝑇̄g (C)
̄𝐡g (W/m2 K)
2400
3600
4600
990
1037
1062
1820
2430
2800
Source: Li (1982).
𝑎 A dished piston running in a 2.5 L engine WOT.
330
Heat and Mass Transfer
300
Piston temperatures (°C)
1 - Center of crown
Figure 11.10 Piston temperature
distribution versus engine speed at
WOT. Adapted from Li (1982).
250
2 - Top ring land
1
3 - Second ring land
2
4 - Middle of skirt
3
200
4
150
100
0
1000
3000
4000
2000
Engine speed (rpm)
5000
gases to lose heat as engine speed increases, whereas the mean heat transfer coefficient
increases because of increased gas motion at higher speeds.
Temperatures in the piston are determined by the average heat flow into the piston
and the effectiveness with which the heat can be dissipated to the oil and the coolant. As
speed increases, the heat flow increases, whereas the overall heat transfer coefficients to
the coolant and oil change little, thus piston temperature increases.
The calculated results show that three areas are particularly important in dissipating
the piston heat input: (1) the ring groove surfaces, (2) the underside of the dome, and
(3) the upper portion of the pin-bearing surface. From the ring grooves, heat flows into the
rings, through the bore, and is eventually absorbed by the coolant. From the underside of
the dome and the surface of the pin bearing, the heat is convected into an air-oil mist and
is eventually absorbed by the oil in the sump.
In zonal modeling of the cylinder gas, the cylinder volume is divided into individual
control volumes, each with its own thermodynamic properties and heat transfer coefficient.
For example, a two-zone model separating the cylinder gases into unburned and burned gas
fractions, with the moving flame separating the two zones is given in Krieger and Borman
(1966). A four-zone model consisting of the central core zone, a squish zone, a head
recess zone, and a piston recess zone was used by Tillock and Martin (1996). With zonal
modeling, the characteristic length and velocity are zone dependent. The characteristic
velocity is usually taken as an effective velocity with components from the mean and
turbulent flow field.
As the number of zones increases to length scales that are much less than the cylinder
bore, the modeling is termed multidimensional. With multidimensional models, the mass,
momentum, and energy conservation equations take the form of partial differential equations, which are solved numerically. Detailed turbulence and reaction rate models are also
required. For example, the use of turbulent heat transfer models in the multidimensional
KIVA code is given in Reitz (1991).
11.6 HEAT TRANSFER CORRELATIONS
Engine heat transfer data can be correlated with the engine thermal conditions using two
nondimensional parameters, the Nusselt and Reynolds numbers. The Nusselt number, 𝑁𝑢,
Heat Transfer Correlations
331
is the ratio of the convection to the conduction heat transfer over the same temperature
difference, expressed in Equation 9.29 as
𝑁𝑢 =
𝐡𝑏
𝑘
(11.26)
where 𝐡 is the heat transfer coefficient, 𝑏 is a length scale, usually the cylinder bore, and
𝑘 is the working fluid thermal conductivity. The Nusselt number is named after Wilhelm
Nusselt (1882--1957), a German engineering professor who made many contributions to
heat transfer, primarily in dimensional analysis, condensation, and heat exchangers.
The Reynolds number, 𝑅𝑒, a ratio of the inertial to viscous fluid forces, is defined in
Equation 11.27:
𝑅𝑒 =
𝜌 𝑈𝑏
𝜇
(11.27)
where 𝜌 is the fluid density, 𝑈 is a characteristic gas velocity, and 𝜇 is the dynamic viscosity.
The characteristic gas velocity in the cylinder depends on a number of parameters,
such as the piston speed, the degree of combustion, the level of turbulence, and the amount
of swirl and tumble present. Since the gas velocity in the cylinder scales with the piston
speed, the mean piston speed is usually chosen as a first order estimate of the characteristic
gas velocity in the cylinder for the Reynolds number. As discussed in Chapter One, the
mean piston speed 𝑈̄ p is
𝑈̄ p = 2 𝑁 𝑠
̄ the
Correlations have been developed for three types of engine heat transfer: (a) 𝑄(𝑥,
̄ 𝜃),
time and spatially averaged engine heat transfer, used in overall energy balance calculations,
(b) 𝑄(𝑥,
̄ 𝜃), the instantaneous spatially averaged cylinder heat transfer, used in engine
performance and heat release analysis, and (c) 𝑄(𝑥, 𝜃), the instantaneous local heat transfer,
typically used in CFD simulations of the detailed combustion and flow fields in the cylinder.
From dimensional analysis, the heat transfer correlations are of the form, Equation
11.28:
𝑁𝑢 = 𝑓 (𝑅𝑒, 𝑃 𝑟) = 𝑎 𝑅𝑒𝑏 𝑃 𝑟𝑐
(11.28)
where 𝑃 𝑟 is the Prandtl number, 𝑃 𝑟 = 𝑣∕𝛼. The Prandtl number is named after Ludwig
Prandtl (1875--1953), a German engineering professor who made many significant contributions to fluid mechanics and aerodynamics.
The convective heat flux 𝑞 ′′ from the cylinder gases to the cylinder wall is
𝑞 ′′ = 𝑄∕𝐴 = 𝐡(𝑇g − 𝑇w )
(11.29)
Combustion gas properties are evaluated at the appropriate mean effective cylinder
gas temperature 𝑇g , which can be obtained using the ideal gas equation for known cylinder pressure and volume. Because of the approximate nature of the correlations, it has
been recommended (Krieger and Borman, 1966) that using air properties for the thermal
conductivity and viscosity of the combustion gases is adequate. Values of the thermal conductivity and dynamic viscosity of air are given in Appendix A. The radiation heat transfer
is included implicitly in the convection correlations or as a stand-alone term, depending
on the computational accuracy desired. For further reading about engine heat transfer, a
comprehensive review is given in Borman and Nishiwaki (1987), and more recently by
Finol and Robinson (2011).
332
Heat and Mass Transfer
Overall Average Heat Transfer Coefficient
A classic correlation for the overall average engine heat transfer coefficient, ℎo , between
the cylinder and the coolant is that of Taylor (1985). The correlation, Equation 11.30 was
developed from engine energy balance data from a variety of engine types -- two- and
four-stroke engines, compression and spark ignition. Typical energy balance measurement
data include engine air and fuel flow rates, coolant flow rate, and temperature rise. The
Nusselt number in the Taylor correlation implicitly includes the conduction and radiation
heat transfer components:
𝑁𝑢 = 10.4 𝑅𝑒0.75
(11.30)
In terms of the mass flow rate into the engine per unit piston area, the Reynolds number
is defined in Equation 11.31 as
𝑅𝑒 =
(𝑚̇ a + 𝑚̇ f ) 𝑏
𝐴 p 𝜇g
(11.31)
The overall heat flux 𝑞 ′′ from an engine cylinder to the coolant is calculated using the
piston area 𝐴 = 𝐴p = 1∕4 𝜋 𝑏2 as a reference area:
𝑞 ′′ =
EXAMPLE 11.1
𝑄̄
= 𝐡o (𝑇̄g − 𝑇c )
𝐴p
Overall Average Heat Transfer Coefficient
Compute the overall average heat transfer coefficient ℎo and heat flux 𝑞 ′′ for a singlecylinder engine with a 0.1 m bore and stroke, average combustion gas temperature of
1000 K, coolant temperature of 350 K, and fuel--airflow rate of 2 × 10−3 kg/s. Assume
𝑘 = 0.06 W/(m K) and 𝜇 = 20 × 10−6 Ns/m2 .
SOLUTION The Reynolds number is
𝑅𝑒b =
(𝑚̇ a + 𝑚̇ f ) 𝑏
= 1274
𝐴 p 𝜇g
The heat transfer coefficient is found using the Taylor correlation:
𝐡o = 10.4 𝑅𝑒0.75
b
𝑘
= 1330 W∕(m2 K)
𝑏
The heat flux from the cylinder to the coolant is therefore
𝑞 ′′ =
𝑄̄
= 𝐡o (𝑇̄g − 𝑇c ) ≈ 0.86 MW∕m2
𝐴p
Instantaneous Cylinder Average Heat Transfer Coefficient
The instantaneous cylinder average heat transfer coefficient, ℎg (𝜃), between the combustion
gas and cylinder wall is a function of crank angle, and is an input to the finite heat
Heat Transfer Correlations
333
release model in this Section represented by Equation 11.39. Since ℎg (𝜃) is a single-zone
cylinder average, properties such as the thermal conductivity and viscosity to be used
in the correlation are the instantaneous spatially averaged values. The instantaneous gas
temperature and density can be determined from the known gas mass and cylinder volume.
Two instantaneous cylinder average heat transfer correlations that have been widely used
are the Annand and the Woschni correlations.
The Annand (1963) correlation was developed from cylinder head thermocouple measurements of instantaneous heat flux. It uses a constant characteristic velocity, the mean
piston speed 𝑈̄ p , and a constant characteristic length, the cylinder diameter 𝑏. The Annand
correlation is
𝑘
𝑞 ′′ = 𝑎1 𝑅𝑒0.7 (𝑇 − 𝑇w ) + 𝑎2 𝜎(𝑇 4 − 𝑇w4 )
(W∕m2 )
(11.32)
𝑏
where 350 < 𝑎1 < 800 depending on the intensity of the charge motion, which was found
to be larger in two-stroke engines than in four-stroke engines. The recommended radiation
term 𝑎2 is equal to 0.58 for diesel combustion and 0.075 for spark ignition engines, and is
relatively small compared with the convection term. The Stefan--Boltzmann constant 𝜎 is
5.67 × 10−8 W/(m2 K4 ).
Another popular correlation for the instantaneous cylinder average heat transfer coefficient is due to Woschni (1967). The Woschni correlation was developed using a heat
balance analysis for each stroke of a direct injection diesel engine and uses a variable
characteristic gas velocity to account for the increased gas velocity induced by combustion.
The Woschni correlation is
𝑁𝑢 = 0.035 𝑅𝑒0.8
(11.33)
The characteristic gas velocity in the Woschni correlation is proportional to the mean
piston speed during intake, compression, and exhaust. During combustion and expansion,
with the valves closed, it is assumed that the gas velocities are increased by the combustion
process, so the characteristic gas velocity has both piston speed and combustion pressure
rise terms:
𝑉 𝑃 − 𝑃m
(11.34)
𝑈 = 2.28 𝑈̄ p + 0.00324 𝑇r d
𝑉r 𝑃r
where
𝑈̄ p = mean piston speed (m/s)
𝑇r = temperature at intake valve closing (K)
𝑉r = cylinder volume at intake valve closing (m3 )
𝑃r = pressure at intake valve closing (kPa)
𝑉d = displacement volume(m3 )
𝑃m = motored pressure (kPa)
The pressure rise due to combustion is the cylinder pressure 𝑃 in the firing engine
minus the cylinder pressure 𝑃m in the motored engine at the same crank angle. The latter
can be estimated by use of the isentropic relation 𝑃m 𝑉m𝛾 = 𝑃r 𝑉r𝛾 = constant, where the
subscript r indicates reference conditions, such as intake valve closing.
The previous equation for the characteristic gas velocity is applicable when the intake
and exhaust valves are closed and combustion is taking place. When the valves are open,
the cylinder gases have a different characteristic velocity resulting from the flow into or
334
Heat and Mass Transfer
out of the cylinder. In this case, the Woschni correlation uses
𝑈 = 6.18 𝑈̄ p
(11.35)
Since thermal conductivity 𝑘 ∼ 𝑇 0.75 , and dynamic viscosity 𝜇 ∼ 𝑇 0.62 , the Woschni
heat transfer coefficient in dimensional form (W/(m2 K)) is given by Equation 11.36:
𝐡g = 3.26𝑃 0.8 𝑈 0.8 𝑏−0.2 𝑇 −0.55
(11.36)
where the units of 𝑃 , 𝑈 , 𝑏, and 𝑇 are in kPa, m/s, m, and K, respectively.
The constants in the Woschni correlation were determined by matching experimental
results from a given engine. When applied to any other engine, the constants for the heat
transfer coefficient and characteristic velocity are estimates at best, and it is not uncommon
to find engineers adjusting them to better match their own engine. Additional correlations
are discussed in Hohenberg (1979).
An illustrative cylinder heat transfer coefficient match is given in Chang et al. (2004).
They modified the Woschni correlation to obtain agreement with measured instantaneous
surface heat flux data in an HCCI engine, especially the effect of engine load. They reduced
the combustion pressure rise term by a factor of six, used the instantaneous chamber height
𝑦 as the characteristic length scale, and changed the temperature exponent to 0.73. Their
correlation is
𝐡g = 3.4𝑃 0.80 𝑈 0.80 𝑦−0.20 𝑇 −0.73
(11.37)
𝑉 𝑃 − 𝑃m
0.00324
𝑈 = 2.28 𝑈̄ p +
𝑇r d
6
𝑉r 𝑃r
(11.38)
where
The finite heat release model introduced in Chapter 2 can now be modified to include
the heat transfer 𝑑𝑄w to the cylinder walls, since we have an expression for the instantaneous
average cylinder heat transfer coefficient ℎg (𝜃). The finite heat release equation, Equation
11.39, with the addition of cylinder wall heat transfer is
[
]
𝑑𝑥
𝑑𝑄w
𝛾 −1
𝑃 𝑑𝑉
𝑑𝑃
=
(11.39)
𝑄in b −
−𝛾
𝑑𝜃
𝑉
𝑑𝜃
𝑑𝜃
𝑉 𝑑𝜃
The heat transfer rate at any crank angle 𝜃 to the exposed cylinder surfaces at an engine
speed 𝑁 is determined with Equation 11.40:
𝑑𝑄w
(11.40)
= 𝐡g (𝜃) 𝐴(𝜃)(𝑇g (𝜃) − 𝑇w )∕(2𝜋𝑁)
𝑑𝜃
The combustion chamber area 𝐴 is a function of crank angle 𝜃, and is the sum of the
combustion chamber area at top dead center 𝐴o and the instantaneous cylinder wall 𝐴w (𝜃)
area. The instantaneous combustion chamber area and volume are thus
𝐴 = 𝐴o + 𝜋 𝑏 𝑦(𝜃)
𝑉 = 𝑉o +
𝜋𝑏2
𝑦(𝜃)
4
or
𝐴 = (𝐴o − 4𝑉o ∕𝑏) + 4𝑉∕𝑏
(11.41)
where 𝑉o is the cylinder volume at top dead center. By reference to Chapter 1, the dimensionless cylinder volume 𝑉̃ (𝜃) = 𝑉 (𝜃)∕𝑉bdc = 𝑉 (𝜃)∕𝑉1 for 𝑙 ≫ 𝑠 is
1 𝑟−1
(1 − cos 𝜃)
𝑉̃ (𝜃) = +
𝑟
2𝑟
(11.42)
Heat Transfer Correlations
335
When the parameters in the heat loss equation are normalized by the conditions at state 1,
bottom dead center, they take the form
𝑄̃ =
𝑄in
𝑃1 𝑉1
ℎ̃ =
4𝐡𝑇1
𝑃1 𝜔𝛽𝑏
𝑄l
𝑃1 𝑉1
(11.43)
4𝑉1
𝑏(𝐴o − 4𝑉o ∕𝑏)
(11.44)
𝑇
𝑇̃ =
𝑇1
𝑄̃ l =
and
𝛽=
The dimensionless heat loss is then
𝑑 𝑄̃ l
̃ + 𝛽 𝑉̃ )(𝑃̃ 𝑉̃ − 𝑇̃w )
(11.45)
= h(1
𝑑𝜃
The original Woschni heat transfer correlation equation, Equation 11.36, is incorporated in the MatlabⓇ program WoschniHeatTransfer.m that is listed in
Appendix F.17. The program is a finite heat release program that can be used to compute
the performance of an engine and calculates the instantaneous heat and mass transfer.
The engine performance is determined by numerically integrating the finite heat release
equation for the pressure, work, heat loss, and cylinder gas mass as a function of crank
angle. The integration starts at bottom dead center (𝜃 = −180◦ ), with initial inlet conditions
given. The integration proceeds degree by degree to top dead center and back to bottom
dead center. Once the pressure and other terms are computed as a function of crank angle,
the overall cycle parameters of network, thermal efficiency, and imep are also computed.
The use of the program is detailed in the following example:
EXAMPLE 11.2
Instantaneous Heat Transfer Coefficient
For the same engine conditions as in Example 2.3 with the start of combustion at
𝜃s = −20◦ , (a) Find the maximum imep and thermal efficiency when cylinder heat transfer
is accounted for using the Woschni heat transfer correlation. (b) Plot the instantaneous heat
transfer coefficient 𝐡g (𝜃), the heat flux 𝑞 ′′ to the cylinder wall, and the cylinder pressure
𝑃 as a function of crank angle.
SOLUTION Assume that the specific heat ratio 𝛾 = 1.3, inlet mixture pressure 𝑃1 = 1 bar, inlet temperature 𝑇1 = 300 K, cylinder wall temperature 𝑇w = 360 K, the top dead center
area/volume ratio 𝐴o ∕𝑉o = 306.6 m−1 , the engine speed 𝜔 = 200 rad/s, and that mass loss
can be neglected. The nondimensional volume parameter 𝛽 and wall temperature 𝑇̃w for
this problem are
𝛽=
(4)(10)
4𝑟
=
= 1.50
𝑏(𝐴o ∕𝑉o ) − 4 (0.1)(306.6) − 4
𝑇̃w = 𝑇w ∕𝑇1 = 360∕300 = 1.2
The given engine parameters are entered into the
program as shown below:
WoschniHeatTransfer.m
function [ ] = WoschniHeatTransfer( )
% Gas cycle heat release code with Woschni heat transfer
336
Heat and Mass Transfer
thetas = -20; % start of heat release (deg)
thetad = 40;
% duration of heat release (deg)
r =10;
% compression ratio
gamma = 1.3;
% gas const
Q = 20;
% dimensionless total heat release
beta = 1.5;
% dimensionless volume
a = 5;
% wiebe parameter a
n = 3;
% wiebe exponent n
omega =200.;
% engine speed rad/s
c = 0;
% mass loss coeff
s = 0.1;
% stroke (m)
b = 0.1;
% bore (m)
T_ bdc = 300;
% temp at bdc (K)
tw = 1.2;
% dimensionless cylinder wall temp
P_bdc = 100;
% pressure at bdc (kPa)
...
The predicted instantaneous heat transfer coefficient 𝐡g (𝜃) is plotted in Figure 11.11. It
rises sharply as heat release commences and has a maximum value of about 2200 W/(m2 K).
The resulting wall heat flux 𝑞 ′′ is presented in Figure 11.12, with a peak heat flux of about
3.6 W/m2 . The cylinder pressure profile and cumulative work and heat loss are plotted in
Figures 11.13 and 11.14. As discussed earlier, the cumulative work is initially negative
due to the piston compression and becomes positive on the expansion stroke. The heat loss
is very small during compression, indicating a nearly isentropic compression process, and
rises rapidly during the heat release process.
Representative thermodynamic parameters are compared with the simple heat release
computation with no heat loss in Table 11.4. With the instantaneous heat transfer included,
the maximum efficiency is reduced from 0.49 to 0.41, and the maximum nondimensional
imep is reduced from 10.96 to 9.11.
2500
2000
1500
1000
500
Figure 11.11 Instantaneous
heat transfer coefficient 𝐡g (𝜃)
(Example 11.2).
0
−150
−100
0
50
Crank angle
−50
(deg)
100
150
Heat Transfer Correlations
337
4
3.5
3
2.5
2
1.5
1
0.5
0
−0.5
−150
Figure 11.12 Heat flux 𝑞 ′′
(Example 11.2).
−100
−50
0
Crank angle
50
(deg)
100
150
50
100
150
100
150
70
Pressure (bar)
60
50
40
30
20
10
Figure 11.13 Cylinder
pressure profile
(Example 11.2).
0
−100
−50
0
Crank angle (deg)
Cumulative work and heat loss
10
Figure 11.14 Cumulative
work and heat loss for
(Example 11.2).
−150
8
Work
Heat loss
6
4
2
0
−2
−4
−150
−100
−50
0
Crank angle
50
(deg)
338
Heat and Mass Transfer
Table 11.4 Comparison of Heat Release Models With and
Without Heat/Mass Loss (Example 11.2)
𝜃s
𝜃d
𝑃max ∕𝑃1
𝜃max
Net work/𝑃1 𝑉1
Heat loss/𝑃1 𝑉1
Efficiency 𝜂
𝜂∕𝜂Otto
Imep/𝑃1
w/ heat loss
w/o heat loss
−20.00
40.00
64.22
11.0
8.20
4.26
0.41
0.82
9.11
−20.00
40.00
67.23
11.0
9.86
0
0.493
0.988
10.96
11.7 HEAT TRANSFER IN THE EXHAUST SYSTEM
Convective heat loss is an important consideration in exhaust pipe and port design, especially for engines with exhaust turbines or catalytic converters. The ports are relatively
short and curved with highly unsteady flow due to valve and piston motion, so the flow will
not be fully developed. The resulting maximum heat flux is of short duration and relatively
high due to the high exhaust gas velocities and temperatures.
Nusselt--Rayleigh number correlations have been developed for the time-averaged heat
transfer in the exhaust system. Malchow et al. (1979) obtained the following correlation,
Equation 11.46 for the average heat loss in a straight circular exhaust pipe with 𝐷∕𝐿 = 0.3,
where 𝐷 is the exhaust pipe diameter and 𝐿 is the pipe length:
𝑁𝑢 = 0.0483 𝑅𝑒0.783
(11.46)
A steady-state correlation for developing flow in a smooth pipe (Incropera and DeWitt,
2007) is
𝑁𝑢 = 0.023 𝑅𝑒0.8 𝑃 𝑟0.33 (1 + 𝐷∕𝐿)0.7
(11.47)
In Equation 11.47, 𝑃 𝑟 is the Prandtl number and 𝐿 is the pipe length. Comparison of the
correlations indicate that the heat transfer in the exhaust pipe is about 50% greater than
would exist in a steady flow in the same pipe.
Hires and Pochmara (1976) correlated experimental results for the instantaneous heat
loss from 10 different exhaust port designs. The suggested correlation equation, Equation
11.48, is
𝑁𝑢 = 0.158 𝑅𝑒0.8
(11.48)
where their Reynolds number is defined as
𝑅𝑒 =
𝑚𝑑
̇
𝜇𝐴
(11.49)
and 𝑚̇ is the instantaneous flow rate, 𝑑 is the throat diameter, 𝐴 is the exit cross section, and
𝜇 is the exhaust gas viscosity. In an exhaust port, it appears that the heat transfer coefficient
is about eight times what it would be in a steady flow in the same port. This is probably
caused by increased turbulence generated by flow separation near the poppet valve.
In that neither Equation 11.46 nor 11.48 includes information about the exhaust valve,
such as lift or valve seat angle, they are applicable to other engines only in so far as they
Radiation Heat Transfer
339
are geometrically similar. As discussed in Chapter 5, at low valve lifts, the port flow is in
the form of a jet, and for larger lifts, the port flow is in the form of developing pipe flow.
The valve geometry and lift have been incorporated into heat transfer correlations by Caton
and Heywood (1981).
11.8 RADIATION HEAT TRANSFER
During the combustion process in an engine, high temperature gases and particulate matter
radiate to the cylinder walls. In a spark ignition engine, the fraction of the gaseous and
particulate matter radiation is very small in comparison to the convection heat transfer
to the cylinder wall. The flame front propagates quickly across the combustion chamber
through a nearly homogeneous fuel--air mixture. Most of the gaseous radiation is in narrow
bands from the H2 , CO2 , and H2 O molecules.
In a compression ignition engine, fuel burns in a turbulent diffusion flame formed
around the fuel spray in the region where the equivalence ratio is close to 1. Soot particles
are formed as an intermediate step in the combustion process. The radiation from the soot
particles during their existence is significant, comprising about 20--40% of the total heat
transfer to the cylinder wall (Dent and Sulaiman, 1977). In contrast to the gaseous radiation,
the soot particles radiate over the entire spectrum. The radiant heat transfer from the flame
will reduce its temperature, which affects the local rate of NO formation. The Woschni
correlation, since it is based on heat flux, includes the radiation heat transfer to the cylinder
wall. The Annand correlation does not include the radiation heat flux.
Radiation heat transfer in participating media, such as the case in a combusting
gas--particulate mixture, is modeled with the radiation transfer equation (RTE). The equation includes the radiant energy absorbed, emitted, and scattered along a given solid angle
direction:
𝜎
𝑑𝐼
= −(𝐾 + 𝜎)𝐼 + 𝐾𝐼b +
𝑃 (Ω, Ω′ )𝐼(Ω′ ) 𝑑Ω′
𝑑𝑠
4𝜋 ∫
(11.50)
where 𝐼 is the radiant intensity in the direction of a solid angle Ω, 𝑠 is the distance in that
direction, 𝐼b is the black body intensity, 𝐾 is an extinction coefficient, 𝜎 is a scattering
coefficient, 𝑃 is the phase function or probability for scattering from solid angle Ω′ into
solid angle Ω. There are a variety of numerical methods for solution of the radiative
transfer equation including flux methods, Monte Carlo techniques, the discrete ordinates
method (DOM), and the discrete transfer method (DTM). The discrete ordinates method
discretizes the radiative transfer equation for a set of finite solid angle directions. The
resulting discrete ordinates equations are solved along the solid angle directions using a
control volume technique.
The radiation transfer equation has been incorporated into multi-dimensional CFD
codes, such as KIVA. Blunsdon et al. (1992) applied the discrete transfer method to the
CFD code KIVA for the simulation of diesel combustion, and also modeled the radiation
from combustion products in a spark ignition engine (Blunsdon et al., 1993). Using the
discrete ordinates method, Abraham and Magi (1997) computed the radiant heat loss in a
diesel engine. Inclusion of the radiant heat loss reduced the peak temperature by about 10%
relative to a nonradiative computation, lowering the predicted frozen NO concentrations.
The exhaust system operates at a temperature high enough so that radiation heat transfer
from the exhaust system to the environment is significant. At full load with a stationary
engine on a test stand, it is possible to make the exhaust system glow red, which indicates
that the radiation emission is in the visible wavelength range. In many engines, a radiation
340
Heat and Mass Transfer
P0 = cylinder pressure
P1 = pressure behind first ring
V1 = volume between first and
second ring
P2 = crankcase pressure
Figure 11.15 Ring pack and 1-D flow
model of blowby.
P0, V0
P2, V2
m 01
m 12
P1, V1
shield is used to reduce the radiation heat transfer from the exhaust manifold to the engine
block and head through the exhaust manifold gasket.
11.9 MASS LOSS OR BLOWBY
There are three primary reasons for an interest in blowby. It influences (1) the gas pressure
acting on the rings that influences the friction and wear characteristics, (2) the indicated
performance, and (3) the hydrocarbon emissions.
Typically, a ring pack consists of two compression rings and an oil ring. The pressure drop across the oil ring is generally negligible. Such a ring pack is represented in
Figure 11.15. A one-dimensional representation of the ring pack is also shown; it consists
of three plenums in series through passages whose sizes are dependent upon the ring gaps,
the piston to cylinder wall clearance, and any ring tilt present. The volumes are all time dependent: 𝑉0 changes because of piston motion; 𝑉1 changes because of ring motion; and 𝑉2
changes because of piston motion (including those of the other cylinders in a multicylinder
engine).
Figure 11.16 shows the results of measurements made for ring gas pressures in a
two-stroke diesel engine (Ruddy, 1979). Notice that about 70◦ after top dead center, the
pressure between the rings 𝑃1 is greater than the cylinder pressure 𝑃o which, if the flows
are quasi-steady, means there is flow of compressed gas from the ring pack back to the
cylinder.
The mass flow from one ring pack plenum to another is governed by the compressible
flow equation, Equation 11.51:
[
(( )
( )(𝛾+1)∕𝛾 )]1∕2
𝑃1
𝑃1 2∕𝛾
2
(11.51)
−
𝑚̇ = 𝜌o 𝐶d 𝐴t 𝑐o
𝛾 −1
𝑃o
𝑃o
The requisite stagnation properties and the specific heat ratio are based on current values in the upstream plenum. The throat area 𝐴t is proportional to the ring gap and the
bore to cylinder clearance. The orifice coefficient 𝐶d , which depends at least on the
Pressure (bar)
Mass Loss or Blowby
341
60
P0 (Cylinder)
50
40
30
(Ring 1)
P1
20
P2
(Ring 2)
10
Figure 11.16 Measured
interring pressure. Adapted
from Ruddy (1979).
240 260 280 300 320 340 tdc
20
40
60
80 100 120 140
Crank angle (deg)
Reynolds number of the flow and also on whether or not the ring is tilted, is not known
with certainty. It is of order unity and its magnitude is fixed by matching between measured and predicted data, typically the average blowby rate and the interring pressure
distribution.
It is typical in such computations to assume that the gas between rings is at a temperature
equal to the average of the piston temperature and the cylinder liner temperature. In so
doing, there is no need to solve the energy equation for each plenum. The equation of
continuity for mass conservation is applied to each plenum where the mass flows in and
out are determined as just described. By simultaneously integrating the resultant ordinary
differential equations, one obtains the mass within each plenum. These are coupled to
equations of motion for each ring, thereby obtaining the plenum volumes. With the plenum
volumes and the mass contained therein, one then uses an equation of state to compute
the pressure in each plenum. A representative computer simulation program used for
computation of ring pack gas flow is RINGPACK (Ricardo, 2014).
It was mentioned that blowby influences the hydrocarbon emissions. During compression and the early stages of combustion, unburned fuel and air is being compressed into
the plenums between the rings. As mentioned, the gases rapidly equilibrate thermally to
the environment and are thus at the average of the piston and liner temperature. In fact, the
heat transfer is so effective that the flame propagating in the cylinder is extinguished when
it tries to propagate into the spaces between the rings. The unburned fuel and air pushed
into the ring pack remains unburned.
Soon after the blowby flow reverses itself due to the decreasing cylinder pressure,
unburned fuel and air emerges from the ring pack back into the cylinder. Since this occurs
late in the expansion stroke, the burned gases in the cylinder are relatively cold, and thus
a large part of this reemerging fuel and air will not be oxidized as it mixed with the
in-cylinder combustion products. Thus, unburned fuel or hydrocarbons will be expelled
from the engine during the exhaust process. Namazian and Heywood (1982) estimated that
anywhere from 2 to 7% of the fuel is wasted in this way. It is interesting to note that this is
another advantage of diesel engines, since in diesel engines, the composition of the cylinder
gas compressed into the ring pack will be primarily air.
342
Heat and Mass Transfer
20
20
Percent
15
[O2]
15
10
10
[CO2]
Figure 11.17 Measured
gas composition at the top
land of a gasoline engine
at WOT. Adapted from
Furuhama and Tateishi
(1972).
5
–60 –30
0
30
Crank angle
60
[CO]
0
–60 –30 0
30
Crank angle
60
Figure 11.17 shows gas concentrations at the top land of a 1.3 L four-cylinder gasoline
engine at WOT at 2000 rpm measured by Furuhama and Tateishi (1972). They mounted
a sampling valve in the piston and opened the valve for 1.25 ms at the same angle in
consecutive cycles. Gases were withdrawn for analysis from the space between the top
land and the cylinder for a variety of crank angles during the cycle.
Their measurements indicated that during the compression stroke, concentrations of
oxygen and hydrocarbons (as 𝑛-hexane equivalent) are high because fuel and air is entering
the ring pack. Likewise, carbon dioxide and carbon monoxide are low in concentration
attributable to the residual gas content. About 15◦ before top dead center, there is a sudden
drop in oxygen and hydrocarbon concentration as burned gases are beginning to enter
the ring pack. About 30◦ after top dead center, it appears that the unburned gases that
entered earlier reemerge, though diluted somewhat by the burned products that entered the
ring pack.
In Chapters 2 and 4, we investigated the influence of blowby on indicated performance.
It was assumed that the flow was always out of the cylinder and that the rate was proportional
to the mass of the cylinder contents. The constant of proportionality 𝐶 was selected so that
about 2.5% of the charge leaked out, consistent with observation. The computations were
deliberately simple for illustration purposes. They ignored the fact that a flow reversal
occurs during the expansion stroke, and thus underestimate the mass of gas that is pushed
into the ring pack during the subsequent compression stroke. As mentioned, when this gas
re-emerges, much of it fails to oxidize and thus fuel is wasted.
11.10 REFERENCES
ABRAHAM, J. and V. MAGI (1997), ‘‘Application of the Discrete Ordinates Method to Compute Radiant
Heat Loss in a Diesel Engine,’’ Numer. Heat Transfer, Part A: Appl., Vol. 31, No. 6, pp. 597--610.
ALKIDAS, A. C. and J. P. MYERS (1982), ‘‘Transient Heat-Flux Measurements in the Combustion
Chamber of a Spark Ignition Engine,’’ ASME J. Heat Transfer, Vol. 104, pp. 62--67.
References
343
ANNAND, W. J. D. (1963), ‘‘Heat Transfer in the Cylinders of a Reciprocating Internal Combustion
Engine,’’ Proc. Instn. Mech. Engrs., Vol. 177, p. 973.
BENDERSKY, D. (1953), ‘‘A Special Thermocouple for Measuring Transient Temperature,’’ Mech.
Eng., Vol. 75, p. 117.
BLUNSDON, C., J. DENT, and W. MALALASEKERA (1993), ‘‘Modeling Infrared Radiation from the
Combustion Products in a Spark Ignition Engine,’’ SAE paper 932699.
BLUNSDON, C., W. MALALASEKERA, and J. DENT (1992), ‘‘Application of the Discrete Transfer Model
of Thermal Radiation in a CFD Simulation of Diesel Engine Combustion and Heat Transfer,’’
SAE paper 922305.
BOHAC, S., D. BAKER, and D. ASSANIS (1996), ‘‘A Global Model for Steady State and Transient S. I.
Engine Heat Transfer Studies,’’ SAE paper 960073.
BORMAN, G. and K. NISHIWAKI (1987), ‘‘Internal Combustion Engine Heat Transfer,’’ Prog. Energy
Combust. Sci., Vol. 13, pp. 1--46.
CATON, J. A. and J. B. HEYWOOD (1981), ‘‘An Experimental and Analytical Study of Heat Transfer in
an Engine Exhaust Port,’’ Int. J. Heat Mass Transfer, Vol. 24 (4), pp. 581--595.
CHANG, J, O. GURALP, Z. FILIPI, D. ASSANIS, T. KUO, P. NAJT, and R. RASK (2004), ‘‘New Heat Transfer
Correlation for an HCCI Engine Derived from Measurements of Instantaneous Surface Heat
Flux,’’ SAE paper 2004-01-2996.
DENT, J. C. and S. J. SULAIMAN (1977), ‘‘Convective and Radiative Heat Transfer in a High Swirl
Direct Injection Diesel Engine,’’ SAE paper 770407.
FINOL, C. and K. ROBINSON (2011), ‘‘Thermal Modeling of Modern Diesel Engines: Proposal of a
New Heat Transfer Coefficient Correlation,’’ Proc. IMechE Part D: J. Automobile Eng., Vol.
225, No. 11, p. 1544--1560.
FURUHAMA, S. and Y. TATEISHI (1972), ‘‘Gases in Piston Top-Land Space of Gasoline Engine,’’ Trans.
Soc. Automotive Eng. of Japan, Vol. 4, pp. 30--39.
HIRES, S. D. and G. L. POCHMARA (1976), ‘‘An Analytical Study of Exhaust Gas Heat Loss in a Piston
Engine Exhaust Port,’’ SAE paper 760767.
HOHENBERG, G. (1979), ‘‘Advanced Approaches for Heat Transfer Calculations ,’’ SAE paper 790825.
INCROPERA, F. , D. DEWITT, T. BERMAN, and A. LAVINE (2007), Fundamentals of Heat and Mass
Transfer, 6th edition, Wiley, New York.
KRIEGER, R. and G. BORMAN (1966), ‘‘The Computation of Apparent Heat Release for Internal
Combustion Engines,’’ ASME Proc. of Diesel Gas Power, paper 66-WA/DPG-4.
LI, C. (1982), ‘‘Piston Thermal Deformation and Friction Considerations,’’ SAE paper 820086.
MALCHOW, G., S. SORENSON, and R. BUCKIUS (1979), ‘‘Heat Transfer in the Straight Section of an
Exhaust Port of a Spark Ignition Engine,’’ SAE paper 790309.
NAMAZIAN, M. and J. B. HEYWOOD (1982), ‘‘Flow in the Piston-Cylinder-Ring Crevices of a SparkIgnition Engine: Effect on Hydrocarbon Emissions, Efficiency and Power,’’ SAE paper 820088.
REITZ, R. (1991), ‘‘Assessment of Wall Heat Transfer Models for Premixed Charge Engine Combustion Computations,’’ SAE 910267.
RINGPACK V.3 Users Manual (2014), Ricardo Software, Inc.
ROBINSON, K., N. CAMPBELL, J. HAWLEY, and D. TILLEY (1999), ‘‘A Review of Precision Engine
Cooling,’’ SAE paper 1999-01-0578.
RUDDY, B. (1979), ‘‘Calculated Inter-Ring Gas Pressures and Their Effect on Ring Pack Lubrication,’’
DAROS Information, Vol. 6, pp. 2--6, Sweden.
RYDER, E. A. (1950), ‘‘Recent Developments in the R-4360 Engine,’’ SAE Quart. Trans., Vol. 4(4),
p. 559.
SHAYLER, P., S. CHRISTIAN, and T. MA (1993), ‘‘A Model for the Investigation of Temperature, Heat
Flow, and Friction Characteristics During Engine Warm-up,’’ SAE paper 931153.
SUN, X., W. WANG, D. LYONS, and X. GAO (1993), ‘‘Experimental Analysis and Performance Improvement of a Single Cylinder Direct Injection Turbocharged Low Heat Rejection Engine,’’
SAE paper 930989.
TAYLOR, C. F. (1985), The Internal Combustion Engine in Theory and Practice, MIT Press, Cambridge,
Massachusetts.
344
Heat and Mass Transfer
TILLOCK, B. and J. MARTIN (1996), ‘‘Measurement and Modeling of Thermal Flows in an Air-Cooled
Engine,’’ SAE paper 961731.
WHITEHOUSE, N. (1970), ‘‘Heat Transfer in Compression-Ignition Engines: First Paper: Heat Transfer
in a Quiescent Chamber Diesel Engine,’’ Proc. Inst. Mech. Engrs., Vol. 185, pp. 963--975.
WISNIEWSKI, T. (1998), ‘‘Experimental Study of Heat Transfer in Exhaust Valves of 4C90 Diesel
Engine,’’ SAE paper 981040.
WOSCHNI, G. (1967), ‘‘A Universally Applicable Equation for the Instantaneous Heat Transfer Coefficient in the Internal Combustion Engine,’’ SAE paper 670931.
11.11 HOMEWORK
11.1 Practical applications of Equation 11.7 are limited because the heat loss to ambient air
is determined by the small differences between much larger numbers. Suppose each term
on the right-hand side can be determined to within ±5%. What tolerances could then be
attached to 𝑄amb ? For nominal values, use the results given in Table 11.1. The most
probable error is computed from the square root of the sum of the squares of the errors of
the RHS terms.
11.2 The average heat flux through a water cooled engine’s cylinder head (𝑘 = 65 W/(m K)) 1.0
cm thick is 0.2 MW/m2 . If the coolant temperature is 85◦ C and the coolant side heat transfer
coefficient is 750 W/(m2 K), what is the average surface temperature on the combustion
chamber and the coolant sides of the cylinder head?
11.3 If an engine cylinder head is changed from iron (𝑘 = 60 W/(m K)) to aluminum (𝑘 = 170
W/(m K)), estimate the change in the heat flux from the head. What is the change in the
gas-side cylinder head temperature?
11.4 At an engine speed of 1000 rpm, what is the approximate penetration depth of the temperature fluctuations in (a) a cast iron block and (b) an aluminum engine block?
11.5 Determine the effect of engine speed on the overall engine heat transfer coefficient of Example 11.1. Plot ℎo versus 𝑁 for 1000 < 𝑁 < 6000 rpm. Assume the volumetric efficiency
𝑒v = 0.9.
11.6 Using the Taylor correlation, Equation 11.30, develop an equation for the overall
heat transfer coefficient ℎo as a function of the engine speed, bore, and volumetric
efficiency.
11.7 With reference to the Taylor correlation, Equation 11.30, the heat transfer coefficient ℎo
increases with engine speed to the 0.75 power, whereas the time available for heat transfer
decreases with engine speed. What is the net effect of increasing engine speed on engine
thermal efficiency?
11.8 Calculate the time-averaged heat transfer (kW) to the coolant from a four-cylinder propanefueled engine with a 120 mm bore and stroke operating at 1200 rpm. The propane fuel
flow rate to the engine is 1.0 g/s with an equivalence ratio 𝜙 = 0.8. The time-averaged
combustion gas temperature is 1200 K, and the coolant temperature is 340 K.
11.9 If 𝑘 ∼ 𝑇 0.75 and 𝜇 ∼ 𝑇 0.62 , how does the heat transfer coefficient of the Woschni correlation, Equation 11.33 vary with pressure and temperature?
11.10 How would you expect the leading coefficient of Equation 11.35 to change if Woschni
had included the effect of the intake valve size?
Homework
345
11.11 A single-cylinder spark ignition cycle engine with a compression ratio 𝑟 = 10 is operated
at full throttle. The initial cylinder pressure and temperature at bottom dead center are 𝑃1
= 1 bar and 𝑇1 = 300 K. The bore and stroke of the engine are 𝑏 = 80 mm and 𝑠 = 90 mm,
with the top dead center area/volume ratio 𝐴o ∕𝑉o = 320 m−1 . The average cylinder wall
temperature 𝑇w = 400 K. The total heat addition, 𝑄in = 2000 J, the start of combustion is at
𝜃s = −15◦ , and the combustion duration 𝜃d = 40◦ . Assume that the ideal gas specific heat
ratio 𝛾 is 1.4, the Wiebe energy release parameters are 𝑎 = 5 and 𝑛 = 3, and the blowby
mass loss coefficient 𝑐 = 0.70.
Using the program WoschniHeatTransfer.m, (a) Find the imep and thermal efficiency. (b) Plot the instantaneous heat transfer coefficient ℎg (𝜃), the heat flux 𝑞 ′′ to the
cylinder wall, and the cylinder pressure 𝑃 as a function of crank angle.
11.12 For the engine in Homework 11.11, if the start of combustion is changed to 𝜃s = −25◦ ,
and the combustion duration 𝜃d = 50◦ , using the program WoschniHeatTransfer.m
program, (a) Find the imep and thermal efficiency. (b) Plot the instantaneous heat transfer
coefficient ℎg (𝜃), the heat flux 𝑞 ′′ to the cylinder wall, and the cylinder pressure 𝑃 as a
function of crank angle.
11.13 What is the average heat transfer coefficient for an exhaust pipe that has an average mass
flow rate of 0.08 kg/s at a mean temperature of 700 K and a pipe diameter of 0.045 m?
Chapter
12
Engine Testing and Control
12.1 INTRODUCTION
The purpose of this chapter is to introduce engine testing, measurement, and control. One
instruments the engine to determine the value of engine parameters such as the engine
torque, engine speed, fuel flow rate, airflow rate, emissions, cylinder pressure, residual
fraction, coolant temperature, oil temperature, and the spark or fuel injection timing. To
measure the performance of an engine one needs to connect the engine to a dynamometer
to control the speed and load.
The testing procedure generally consists of operating the engine at different speeds
and loads and measuring the parameters of interest for a given test. An energy balance can
also be performed as a check to determine the various energy flow paths in the engine,
that is, to the dynamometer, coolant, ambient, and exhaust. Exhaust gas measurement and
analysis are performed to determine the emissions produced by the engine.
Some measurements are rather straightforward and require little, if any, explanation.
The coolant temperature is easily measured by insertion of a thermocouple or thermistor
into the coolant. Some of the measurements require analysis to obtain the desired result. For
example, the air--fuel equivalence ratio is determined from the combustion equations and
measurements of the composition of the exhaust gases. The testing results are normalized
into parameters such as the specific fuel consumption or the mean effective pressure.
The internal combustion engine is a complex electromechanical system with a number
of embedded sensing and control systems. These systems include air--fuel ratio, spark
timing, knock, idle speed, and exhaust gas recirculation (EGR) control. There is a need
to provide real-time information about the engine state to an engine control system, and
subsequently the control system needs to be able to change the operating state of the engine.
The two primary inputs to the control system for engine testing are the engine speed
and engine torque. The control parameters for a spark ignition engine are the spark timing,
valve timing, exhaust gas recirculation, and fuel injector flow. With a spark ignition automobile engine, the air--fuel ratio is tightly controlled to stoichiometric conditions to ensure
proper performance of the exhaust three-way catalytic converter, and the ignition timing is
controlled to prevent knock. For a compression ignition engine, major control parameters
are the fuel injector flow rate and the injection timing.
Sensors for additional engine parameters are in development for use in production vehicles. These include an optical combustion sensor to detect the peak combustion pressure,
that is, peak torque. Similarly, a crankshaft torque sensor is also under development. These
two sensors can be used to maintain the engine at maximum brake torque conditions. In the
emissions area, a vehicle NO𝑥 sensor is being developed to allow an engine to be operated
with closed-loop control of the NO𝑥 levels.
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
346
Instrumentation
347
Figure 12.1 Engine on
hydraulic dynamometer
test stand. (Courtesy Land
& Sea, Inc.)
12.2 INSTRUMENTATION
Dynamometers
The dynamometer is a device that provides an external load on the engine and absorbs
the power produced by the engine, as shown in Figure 12.1. The earliest dynamometers
were brakes that used mechanical friction to absorb the engine power, hence the power
absorbed was called the ‘‘brake horsepower.’’ The types of dynamometers currently used
are hydraulic or electric. A hydraulic dynamometer or water brake is constructed of a vaned
rotor mounted in a casing mounted to the rotating engine shaft. A continuous flow of water
is maintained through the casing. The power absorbed by the rotor is dissipated in fluid
friction as the rotor shears through the water. Adjusting the level of water in the casing
varies the torque absorbed.
There are a number of different kinds of electric dynamometers. These include direct
current, regenerative alternating current, and eddy current. The power absorbed in an electric
dynamometer is converted into electrical energy, either as power or eddy currents. The
electricity can then be dissipated as heat by resistance heating and transferred to a cooling
water or air stream. In direct current or regenerative alternating-current machines, the
electricity generated can be used, and transformers are available that allow it to supplement
a power system. Historically, direct-current machines have offered the greatest testing
flexibility but at the greatest cost.
Engine dynamometers can further be classified depending upon whether or not they
also have the capability to motor an engine, that is, spin an engine not producing power,
similar to the operation of an electric the starter motor of an automobile engine. Hydraulic
dynamometers cannot motor an engine. Strictly speaking, neither can eddy-current machines, but because they are often configured into a package with an electric motor to run
an engine, for practical purposes the distinction is moot.
The method most commonly employed to measure torque is shown in Figure 12.2.
The dynamometer is supported by trunnion bearings and restrained from rotation only by a
348
Engine Testing and Control
Dynamometer
R2
R0
R1
Bearing
Strut
Figure 12.2 Torque
measurement using a
cradle-mounted dynamometer.
Load
cell
strut connected to a load cell. Whether the dynamometer is absorbing or providing power,
a reaction torque is applied to the dynamometer. Hence, if the force applied by the strut is
𝐹 , then the torque applied to the engine is given by Equation 12.1:
𝜏 = 𝐹 𝑅o
(12.1)
where 𝑅o is defined in Figure 12.2. The load cell measures the force 𝐹 . For calibration,
lever arms are located at 𝑅1 and 𝑅2 for hanging known weights.
Since the work done in rotating the engine’s crankshaft through one revolution, or 2𝜋
radians, is 2𝜋𝜏, it follows that the mean effective pressure for two-stroke engines is equal
to 2𝜋𝜏∕𝑉d and the mean effective pressure for four-stroke engines is equal to 4𝜋𝜏∕𝑉d ,
respectively.
If the engine is absorbing energy, then the brake mean effective pressure (bmep) is
determined. If the engine is being motored, then the motoring mean effective pressure
(mmep) is determined (which, as explained in Chapter 10, is an approximate measure of
the friction losses in an engine).
With an appropriate control system, the dynamometer can be used to control either
engine speed or torque. For control of engine speed, the dynamometer applies whatever
load is required to maintain that speed. For example, if the engine being tested were a
spark ignition engine, then the response of the dynamometer to the operator increasing the
inlet manifold pressure by opening the throttle would be to increase the load (resistance to
turning or applied torque) to maintain the speed.
With torque control selected, the dynamometer maintains a fixed load. For example,
the dynamometers response to opening the throttle of a spark ignition engine would be
to maintain a constant applied torque. In this case, the engine speed would increase to a
point where the friction mean effective pressure in the engine would have increased by an
amount equal to the increase in the net indicated mean effective pressure.
Since about one third of the input fuel energy to the engine ends up as heat transfer
to the coolant, a cooling tower or radiator is required for the dynamometer stand. The
cooling tower will control the coolant temperature. A complete test stand has, in addition,
provisions to control the fuel and air temperature, the atmospheric pressure, and the air
humidity.
Instrumentation
349
Crank Angle
The crankshaft position is used by the engine control system for ignition timing control,
that is, spark advance, which is defined as degrees before top dead center. The crankshaft
position can be determined from measurements made in a number of locations on the
mechanical drive train, for example, on the crankshaft, camshaft, or distributor shaft.
Noncontacting methods are used, which are usually electrical, but optical methods have
also been devised.
A Hall effect sensor is commonly used on the camshaft or distributor shaft. The Hall
effect, discovered in 1879 by E. H. Hall (1855--1938), an American physicist, is due to
electromagnetic forces acting on electrons in metals and semiconductors. If a current is
passed through a semiconductor that is placed near a magnet, a voltage is developed across
the semiconductor perpendicular to the direction of current flow 𝑉⃗ and perpendicular
⃗ The voltage results from the Lorentz force (𝑉⃗ × 𝐵⃗
to the direction of magnetic flux 𝐵.
acting on the electrons in the semiconductor. The Lorentz force is named after H. Lorentz
(1853--1928), a Dutch physicist. The voltage is proportional to the magnetic flux so that if
the magnetic flux is changed, the voltage will change. There are a number of Hall effect
sensor configurations. With a shielded field sensor, tabs are placed on a rotating disc,
mounted on the distributor shaft, and the Hall sensor and magnet are placed on opposite
sides of the tab. Each time the tab passes between the magnet and the Hall sensor, the
magnetic reluctance of the tab will decrease the magnetic flux intensity at the sensor that
causes a corresponding decrease in the sensor Hall voltage. The voltage is independent of
engine speed, so the Hall effect sensor can be used even if the engine is not running.
Engine Speed
The engine speed is measured with optical or electrical techniques. One optical technique
uses a disk with holes mounted on the revolving engine shaft. A light-emitting diode is
mounted on one side of the disk and a phototransistor is mounted on the other side. Each
time a hole on the disk passes by the optical sensor, a pulse of light impinges on the
phototransistor that generates a periodic signal, the frequency of which is proportional to
engine speed.
Many engines use a notch in the flywheel and a magnetic reluctance sensor. The
engine speed is found by measuring the frequency at which the notch passes by the position
sensor. The sensor is an electromagnet whose induced voltage varies with the change in
the magnetic flux. As the notch in the flywheel passes by the sensor, the induced voltage
will first decrease, then increase. If the engine is not running, there will be no change in
the magnetic flux, and the magnetic reluctance sensor will not produce any voltage.
Fuel Flow Measurement
An old, but accurate and simple, way to measure the cumulative fuel flow to an engine is
to locate the fuel supply on a weighing bridge and time the period required to consume
a certain weight of fuel. The essence of such a system is shown in Figure 12.3. This
method works equally well for both liquid and gaseous fuels. For liquid fuels, a pipette
and stopwatch can be used, a method used to calibrate fuel flow meters. A small positive
displacement turbine can be installed in the fuel line as an electronic fuel flow transducer.
Basically, the rotational speed of the turbine is proportional to the fuel flow rate. These
transducers are also convenient in terms of minimizing bulk in the test cell, maximizing
safety, and maintaining a clean fuel system. Unfortunately, they measure a volumetric
350
Engine Testing and Control
Scale
Beam
Figure 12.3 Fuel flow
measurement using a
weighing bridge. Adapted
from Lynch and Smith (1997).
Fuel
tank
Balance
weight
Fluid
force
Flow
Figure 12.4 Coriolis
effect flow meter.
(a) Sensor tube
vibration. (b) Forces
acting on sensor
tube in upward motion.
(c) End view indicating
the force couple and
tube twist. (Courtesy
Micro Motion, Inc.)
Fluid
force
(a)
Fluid
force
(b)
Twist
angle
Twist
angle
(c)
Fluid
force
flow rate instead of a mass flow rate, and the calibration is weakly dependent upon the
fuel viscosity. Thus, in practice, calibration curves have to be established as a function
of the fuel temperature (and possibly pressure) and new ones generated if the fuel type
is varied. The calibration curve needs to span the nominal range of fuel flow rates that
can be as large as 50 : 1. At considerably greater cost than the turbine-type flow meters,
there are other types, such as the ‘‘Flowtron’’ meter, and the Coriolis flow meter. The
‘‘Flowtron’’ meter is the hydraulic equivalent of the Wheatstone bridge circuit. The bridge
comprises four matched orifices and a recirculating pump. The external fuel flow through
the meter generates a pressure imbalance that is proportional to the mass flow rate. Coriolis
flow meters pass the fuel flow through a vibrating U-tube, as shown in Figure 12.4. The
Coriolis force (2𝜔̄ × 𝑈̄ ) acting on the flow will generate a twisting moment in the tube
since the flow reverses direction as it passes through the U-tube. A strain gage mounted
on the tube measures the magnitude of the twist that is proportional to the flow rate. The
accuracy and repeatability of the Flowtron and Coriolis meters is excellent, better than 0.5%
or less.
The return flow from fuel injection systems needs to be taken into account in fuel flow
measurements, since it is of the same order of magnitude as the fuel flow through the fuel
injectors. One approach is to cool the return fuel to the temperature of the supply fuel and
connect it to the supply fuel downstream of the fuel flow meter.
Airflow Measurement
The inlet air mass flow rate in vehicular engines is measured by a constant temperature hot
film anemometer. The principle of operation is very elegant. A small resistive wire or film
Instrumentation
351
To
Steady flow
air meter
Inlet
Exhaust
Surge
tank
Po
Figure 12.5 Single-cylinder
engine equipped with inlet
surge tank and steady
flow air meter.
placed in the airflow is heated by an electric current. The current required to maintain the
film at a constant temperature above the ambient is proportional to the mass flow rate of
the air. The sensor is placed at one leg of a Wheatstone bridge circuit so that the proper
current is sent through the sensor to maintain a constant sensor temperature.
Airflow to engines cannot be measured with the same precision as the fuel flow. There
are two main reasons for this: (1) the instrumentation available is at best accurate to within
about 1%, and (2) it is harder to ensure that all of the air delivered to the engine is metered
or retained. Air can leak into and out of the cylinder of engines, for example, through the
valve guides. As a result, as recommended by Stone (1989), it is wise to measure airflow
not only directly but also indirectly using exhaust gas analysis. A discussion of airflow
measurement via exhaust gas analysis is given later in this chapter.
A common problem in measuring airflow is that the flow is unsteady or periodic;
however, the available meters are usable only in a steady flow. A similar problem can
be encountered in measuring fuel flow. The severity of the problem decreases with an
increasing number of cylinders sharing a common intake manifold and with an increase in
the volume between the meter and the intake ports. The volume acts as a fluid capacitor to
damp out fluctuations at the meter. The flow at the meter is smoother with multicylinder
engines than with single-cylinder engines because the cylinders are out of phase with
one another; thus, as the peaks and valleys in the flow rates to individual cylinders are
superimposed, the flow at the meter becomes smooth.
A solution for the worst case, that is, for a single-cylinder engine, that can be applied
to steady-state engine testing is illustrated in Figure 12.5. All of the air to be delivered
to the engine is metered by a steady-state airflow meter located upstream of a surge tank.
Kastner (1947) recommends that the volume of the surge tank be at least about 250 times
the displacement volume of the engine. The various types of airflow meters that can be
used include the following:
ASME Orifice Often employed as a secondary standard to calibrate other meters; flow
rate depends on the square root of the pressure drop across the orifice, so a range of
orifice sizes is used to cover the airflow rate range.
Laminar Flow Meter A bundle of tubes (not necessarily round in cross section) sized so
that the Reynolds number in each is well within the laminar regime; flow rate depends
linearly on the pressure drop across the meter.
352
Engine Testing and Control
Figure 12.6 Various air flow meters and associated pressure/temperature measurements.
Critical Flow Nozzle A Venturi in which the flow is choked; the flow rate is then linearly
dependent upon the delivery pressure (an external compressor is thus required) and
independent of the pressure in the surge tank.
Turbine Meter The airflow rate is linearly dependent upon the rotational speed of the
turbine.
Hot Wire Meter A hot wire anemometer is inserted into the flow to measure the centerline velocity; the airflow rate is proportional to center-line velocity.
No matter which of the various methods is used, measurements of temperature and pressure
also have to be made. Key components of systems employed using these various meters
are identified in Figure 12.6. The calibration coefficients of the meters are a function of
the Reynolds number of the flow through the meters. For transient engine testing, only the
hot wire meter can be used, as it can measure the instantaneous mass flow rate; it can also
be used in steady-state testing where the required air box is viewed as a nuisance. It must,
however, be used with care because it is possible that in some engines a flow reversal will
occur and the meter does not know whether the flow is going forward or backward.
Correction factors are used to adjust measured data to standard atmospheric temperature
and pressure conditions. The specific correction procedures are usually included in the
laboratory practice manual.
Injector voltage
Instrumentation
353
Pulse interval
Open
Closed
Injector voltage
Time
(a)
Figure 12.7 Fuel injector
control voltage. (a) Part load.
(b) Full load.
Pulse interval
Open
Closed
Time
(b)
Air--Fuel Ratio
The fuel flow rate is computed from the air mass flow rate and the requirement that a spark
ignition engine maintains a stoichiometric air--fuel ratio, so the fuel flow rate is directly
proportional to the airflow rate. The fuel metering of a fuel injector is performed by a
solenoid-operated plunger attached to a needle valve. The plunger is normally closed, so
that when the solenoid is not energized, no fuel can flow through the fuel injector. When
the engine control unit energizes the solenoid, the valve is lifted and allows fuel to flow
through the injector nozzle into the intake port.
The fuel pressure is regulated to a fixed value, so the amount of fuel injected is
proportional to the time that the valve is open. The proportionality constant for fuel flow
and injector pulse width is determined experimentally. The fuel injector control voltage is
pulse-width modulated, as shown in Figure 12.7. The ECU uses a torque-based mean value
model to control the pulse width. The width of the solenoid voltage pulse from the ECU
depends on the engine load, and the frequency is proportional to engine speed.
Manifold and Ambient Air Pressure
The air pressure in the intake manifold is used by the engine control system as an indication
of engine load. Higher manifold pressures correspond to higher loads, since as the throttle
is opened to meet an increasing load, the pressure drop across the throttle decreases, and the
manifold pressure increases. The manifold air pressure is measured by the displacement of
a diaphragm that is deflected by manifold pressure. There are several types of diaphragm
sensors. A common one is the silicon-diffused strain gauge sensor. This sensor is a thin
square silicon diaphragm with sensing resistors at each edge. One side of the diaphragm is
sealed under vacuum, and the other side is exposed to the manifold pressure. The resistors
are piezoresistive so that their resistivity is proportional to the strain of the diaphragm. A
Wheatstone bridge circuit is used to convert the resistance change to a voltage signal. The
pressure fluctuations from the finite opening and closing of the intake valves are filtered
out with a small diameter vacuum hose so that the time-average pressure is measured.
354
Engine Testing and Control
Throttle Position
The throttle position is a control system input as it is controlled by the engine operator.
The throttle position is measured by a variable resistor or potentiometer attached to the
axis of the sensor butterfly throttle valve. As the throttle rotates, the internal resistance
of the sensor is changed proportional to the throttle angle change. For a spark ignition
engine, the airflow rate into the intake manifold is governed by the throttle position, that
is, the throttle functions as a mass airflow control valve. The airflow rate is a nonlinear
function of throttle position, and for relatively small throttle angle changes, the airflow
versus throttle angle relationship is linearized in control modeling.
Exhaust Gas Recirculation
Exhaust gas recirculation is used to reduce peak combustion temperatures and thus NO𝑥
emission levels. The exhaust gas recirculation actuator is a vacuum-operated, spring-loaded
diaphragm valve. The amount of EGR depends on the combustion stability limits, engine
speed, and temperature. The EGR valve is normally closed. When EGR is needed, the
engine control system energizes a solenoid valve to supply vacuum to the diaphragm valve.
The diaphragm valve opens to allow exhaust gases to flow into the intake manifold and
mix with the incoming fuel--air mixture.
Inlet Air and Coolant Temperature
The inlet air temperature and coolant temperature are measured with thermistors. The
thermistors are mounted in a housing placed in the fluid stream. The coolant temperature
is used to indicate engine warm-up and overheating states.
12.3 COMBUSTION ANALYSIS
A number of methods have been used to measure pressure as a function of cylinder volume
for combustion analysis purposes. We will restrict our attention to piezoelectric transducers,
since they are the method preferred by most engine laboratories.
The piezoelectric effect is the generation of an electric charge on a solid by a change
in pressure. Consider that a crystalline solid is made up of positive and negative charges
distributed over a space in a lattice structure. If the distribution of charges is nonsymmetrical,
stressing the crystal will distort the lattice and displace positive charges relative to negative
charges. A surface that was electrically neutral may become positive or negative. Substances
such as salt (NaCl) have a symmetrical distribution of charges and therefore stresses do
not lead to piezoelectricity. There are two primary piezoelectric effects: (1) the transversal
effect in which charges on the 𝑥-planes of the crystal result from forces acting upon the
𝑦-plane, and (2) the longitudinal effect in which charges on the 𝑥-planes of the crystal result
from forces acting upon the 𝑥-plane. In Figure 12.8(a), an example of the transversal effect,
the quartz is cut as a cylinder with two 180◦ or three 120◦ sectors. The potential difference
between the outer and inner curved surfaces of the cylinder is a measure of the gas pressure.
In Figure 12.8(b), an example of the longitudinal effect, the quartz is cut into a number of
wafers electrically connected in parallel. The potential difference is measured between the
plane surfaces.
Piezoelectric transducers can be obtained with internal coolant passages and with a
temperature compensator. Note that a rise in temperature will cause the housing to expand
Combustion Analysis
355
Receptacle
Housing
Preload sleeve
Charge pick-up
Temperature insulator
(a)
Diaphragm
Electrical contact
Coolant
passageway
Insulator
Coolant passageway
(water)
Electrode
(Hermetically
welded)
Figure 12.8 Quartz
piezoelectric pressure
transducers. (a) (Courtesy
Kistler Instrument, Inc.)
(b) Courtesy AVL Corp.
Housing
Quartz wafers
Temperature
compensator
Sensing
surface
Diaphragm
(b)
and thereby relieve the precompressed crystals from load. Piezoelectric transducers can
also be obtained with flame shields to reduce flame impingement errors. Such errors are
also reduced by coating the diaphragm with a silicone rubber to act as a heat shield.
Quantitative use of piezoelectric transducers is nontrivial. Care and methodical procedure is required. The reader who is using these transducers is advised to also consult
classic SAE papers by Brown (1967), Lancaster et al. (1975), and Randolph (1990), as well
as to follow the manufacturer’s calibration procedures methodically. A typical system for
measuring cylinder pressure as a function of cylinder volume is shown in Figure 12.9. A
crank angle encoder is used to establish the top dead center position and the phasing of
cylinder pressure to crank angle.
Computer-based combustion analysis hardware and software are used to acquire and
analyze the pressure data. The hardware consists of high-speed A/D data acquisition systems
Engine Testing and Control
Transducer signal
Top dead center
Transducer
Charge
amplifier
Computer
tdc
slot
Figure 12.9
Schematic of pressure
measurement system.
Light
detector
PC-based data
acquisition and
analysis system
Light detector
signal
Light
source
Flywheelmounted
chopper
800
700
600
P (psi)
356
500
400
300
200
Figure 12.10 Representative
cylinder pressure versus
crank angle.
100
0
–180
–90
TDC
90
180
Crank angle (deg)
and dedicated digital signal processors. The software performs statistical and thermodynamic analysis of the pressure data in real time. To study the effect of cycle-to-cycle
variation, the analysis can be performed on an individual cycle and also on an ensemble
average of many cycles. Measurements of cylinder pressure can be used to determine not
only the location of peak pressure but also the instantaneous heat release, burn fraction,
and gas temperature. For additional information, the reader is referred to papers by Foster
(1985) and Cheung and Haywood (1993). Representative pressure versus crank angle data
are plotted in Figure 12.10. Using the known slider--crank geometry, the pressure data can
be plotted as a function of cylinder volume, see Figure 12.11, and in the form log 𝑃 versus
log V, see Figure 12.12. The nonreacting portions of the compression and expansion strokes
are modeled as polytropic processes where 𝑃 𝑉 𝑛 = constant. The polytropic exponent 𝑛
can be found from the slope of the curve on the log 𝑃 versus log 𝑉 plot. Note that the
intake and exhaust pumping loop characteristics and the pressure sensor fluctuations are
much more evident when the pressure data are plotted on logarithmic coordinates.
The instantaneous heat release, see Figure 12.13, is deduced from the cylinder pressure measurements through the use of the differential energy equation, Equation (2.34),
rearranged in this chapter as Equation 12.2, where 𝑑𝑄wall is the heat transfer to the wall.
𝑑𝑄wall
𝛾
𝑑𝑄
1
𝑑𝑃
𝑑𝑉
=
𝑉
+
𝑃
+
𝑑𝜃
𝛾 − 1 𝑑𝜃
𝛾 − 1 𝑑𝜃
𝑑𝜃
(12.2)
The integral of the instantaneous heat release provides the burn fraction curve, shown
in Figure 12.14. The average gas temperature can be computed from the measured pressure
Combustion Analysis
357
800
700
P (psi)
600
500
400
300
200
100
0
Figure 12.11 Cylinder
pressure versus volume.
TDC 30
60
90
120
60
90
150 BDC
log P (psi)
V
500
300
200
100
50
30
20
10
0
Figure 12.12 Log cylinder
pressure versus log volume.
TDC
30
120 BDC
log V
BTU/CA
0.04
0.03
dQ
0.02
0.01
0
Figure 12.13 Instantaneous heat release
versus log volume.
–60
–30
TDC
30
60
90
60
90
Crank angle (deg)
1.0
0.9
0.8
xb 0.6
0.4
0.2
Figure 12.14 Cumulative
heat release versus crank
angle.
0
–60
–30
TDC
30
Crank angle (deg)
358
Engine Testing and Control
Tmax
Tcylinder (°F)
3000
Figure 12.15 Cylinder
average temperature
versus crank angle.
(ic: intake valve close and
eo: exhaust valve open.)
Teo
2000
1000
Tic
0
–180
–90
TDC
90
180
Crank angle (deg)
and the ideal gas equation, Equation 12.22. The temperature shown in Figure 12.15 is an
average of the burned and unburned gas temperatures. The mixture mass 𝑚 is evaluated
from the conditions at a convenient reference, such as intake valve closing.
12.4 EXHAUST GAS ANALYSIS
Electronic instruments that are easy to use and reliable are available for several of the
exhaust constituents of interest to us including carbon dioxide, carbon monoxide, hydrocarbons, oxygen, and nitrogen oxides. Many laboratories, especially those studying or
testing emissions, have a set of these instruments mounted together with a suitable sample
handling system. We will briefly explain how the instruments for the species mentioned
operate, and then we will look at how experimental data can be used to compute a fuel--air
ratio.
Carbon Dioxide and Carbon Monoxide
Nondispersive infrared analyzers (NDIR) are used for carbon dioxide and carbon monoxide.
They can also be used for methane, hexane, nitric oxide, sulfur dioxide, ethylene, and water.
The principle of operation of the infrared analyzer is based on the infrared absorption
spectrum of gases. For the most part, gases are transparent to electromagnetic radiation.
However, at certain frequencies in the infrared spectrum, the energy associated with a
photon coincides with that required to change a molecule from one quantized energy level
to another. At those frequencies, a gas will absorb radiation. The concentration of a given
compound in a gas mixture can then be determined from the absorption characteristics. As
shown in Figure 12.16, carbon dioxide absorbs at about 4.2 µm; whereas carbon monoxide
absorbs at about 4.6 µm. Thus, by using a radiation analyzer with a sensitivity as shown,
one can detect carbon dioxide in a sample without interference from any carbon monoxide
that may also be present.
The operation of an infrared analyzer is shown in Figure 12.17. The analyzer passes
infrared radiation through two cells: one a reference cell containing a nonabsorbing background gas and the other a sample cell containing a continuous flowing sample. The detector
is filled with the component gas of interest to absorb infrared radiation transmitted through
the two cells. The detector will absorb less radiation on the right than on the left because
of the attenuation in the sample cell causing a diaphragm to deflect in proportion to the
difference in the rates of energy absorption. Since the deflection will depend on the component density in the sample stream, the amount of deflection can be sensed and displayed
359
Exhaust Gas Analysis
Percent transmittance
100
80
CO2 transmission
60
40
20
CO2 detector
sensitivity
CO transmission
Figure 12.16 CO and CO2
infrared transmittance
spectra.
0
4
5
Wavelength (m)
Infrared
source
Chopper
Sample in
Sample
cell
Reference
cell
Sample out
Detector
Diaphragm
distended
Figure 12.17 Infrared analyzer operation.
(Courtesy Beckman Instruments, Inc.)
Control unit
Component of interest
Other molecules
on an electric meter calibrated to read in units of concentration. Notice that by filling the
detector with the component of interest, one automatically obtains the desired sensitivity
so as to eliminate interference from other components.
Hydrocarbons
Hydrocarbon detection is performed with a flame ionization detector (FID). Introduction
of hydrocarbon molecules into a hydrogen--air flame produces, in a complex process,
electrons and positive ions. By burning a sample of the exhaust gas in an electric field,
positive ions are produced in an amount proportional to the number of carbon atoms
introduced into the flame. A schematic of such a burner is shown in Figure 12.18. The
sample is mixed with the hydrogen and diluent fuel and burned in a diffusion flame. The
combustion products pass between electrodes producing an ion current. The hydrocarbon
concentration is proportional to the ion current.
360
Engine Testing and Control
Ion current
+
–
–
Cathode
+
+
+
–
–
Flame
Fuel inlet
Figure 12.18 Schematic of a flame
ionization detector (FID).
+
Anode
–
Electric field
–
Air inlet
Sample inlet
Table 12.1 FID Characteristic Response
Molecular structure
Methane
Alkanes
Aromatics
Alkenes
Alkynes
Carbonyl radical
Nitrile radical
Approximate response
1.0
1.0
1.0
0.95
1.3
0
0.3
The magnitude of the current depends somewhat on the molecular structure of the
hydrocarbon being detected. The characteristic response of a given molecular structure
normalized by the response to methane is given in Table 12.1.
According to Table 12.1, the following concentrations would all read approximately
1% on the meter:
1.00% of CH4 , methane;
0.1% of C10 H22 , decane;
0.132% of C8 H16 , octene; and
0.385% of C2 H2 , acetylene.
The flame ionization detector gives no information about the type of hydrocarbons in the
exhaust or their average hydrogen to carbon ratio. In recognition of this latter point, it is
preferred to report measurements as ppmC (particles per million carbon) rather than as ppm
CH4 or C3 H8 or C𝛼 H𝛽 equivalent.
Hydrocarbon speciation is performed with a Fourier transform infrared analyzer
(FTIR), a gas chromatograph, or a mass spectrometer. The FTIR operates on the same
principle as the NDIR but also computes a Fourier transform of the infrared absorption
Exhaust Gas Analysis
361
Photomultiplier
Window
Photons
Sample in
Figure 12.19 Model
representation of the
reactor in a chemiluminescence nitric oxide reactor.
Ozone in
Fan to mix
reactants
Products out
Reactor
volume V
spectrum of the gas mixture. It is useful for the detection of methanol and formaldehyde. A
gas chromatograph uses a solid or solid--liquid column to separate the hydrocarbon species.
The detection limits for gas chromatographs used in conjunction with a flame ionization
detector are on the order of 10 parts per billion carbon (ppbC).
Nitrogen Oxides
Nitrogen oxides (NO𝑥 ) are measured with a chemiluminescence detector (CLD). Chemiluminescence is the process of photon emission during a chemical reaction. When nitric
oxide (NO) reacts with ozone (O3 ), chemiluminescence from an intermediate product nitric
dioxide (NO2 ) occurs during the reaction. The amount of nitric oxide present is proportional
to the number of photons produced.
A chemiluminescence reactor model is shown in Figure 12.19. The exhaust gas sample
is first passed through a catalyst to convert nitric dioxide (NO2 ) to nitric oxide (NO) prior
to delivery to the reactor. The reactor has an exhaust gas sample port, an ozone inlet port,
and an outlet port. The photons produced are measured with a photomultiplier. An optical
filter is used to filter out photons from non-NO2 chemiluminescence reactions that produce
photons outside the wavelength band between 0.60 and 0.66 µm.
To simplify the reaction analysis, the reactor is assumed to be perfectly stirred, so
the concentration of reactants is uniform throughout the reactor. The chemical reactions
involved in this process are
NO + O3 → NO∗2 + O2
(12.3)
NO∗2 → NO2 + photon
(12.4)
NO∗2 + M → NO2 + M
(12.5)
The asterisk in the above equations denotes NO2 in an electronically excited state and
M is a symbol chemists use to denote any molecule in the system. The nitric oxide (NO)
reacts with ozone O3 to produce electronically excited nitric dioxide NO∗2 . The excited
nitric oxides can be deactivated by emission of a photon or by collision with any other
molecule.
The conservation equation for excited nitric dioxide is
𝑑
[NO∗2 ] = 𝑘1 [NO][O3 ] − 𝑘2 [NO∗2 ] − 𝑘3 [NO∗2 ][M] − 𝑉̇ 𝑓 [NO∗2 ] = 0
𝑑𝑡
(12.6)
362
Engine Testing and Control
Equation 12.6 indicates that the rate at which excited nitric dioxide NO∗2 is produced
by reaction with ozone in the reactor is balanced by the reduction rates due to photon
emission, molecular collision, and the rate at which it flows out of the reactor. The braces
in the conservation equation denote the concentrations in units of mol/m3 and 𝑉̇ f is the
volumetric flow rate of products leaving the reactor. The 𝑘’s are the rate coefficients for
the three reactions, Equations 12.3--12.5. When the system is in steady state, the rate of
change of concentration of any species in the reactor is zero. The steady-state concentration
of NO∗2 is therefore
[NO∗2 ] =
𝑘1 [NO][O3 ]
𝑘2 + 𝑘3 [M] + 𝑉̇ 𝑓
(12.7)
At steady state, the rate at which photons leave the system is equal to the rate at which
they are produced in reaction 12.4. Therefore, the photon intensity, 𝐼, measured by the
photomultiplier is
𝐼 = 𝑘2 [NO∗2 ] =
𝑘2 𝑘1 [NO][O3 ]
𝑘2 + 𝑘3 [M] + 𝑉̇ 𝑓
(12.8)
The rate of change of concentration of nitric oxide is assumed to be zero, so
𝑑
[NO] = 𝑘1 [NO][O3 ] − 𝑉̇ 𝑓 [NO]sample = 0
𝑑𝑡
(12.9)
Upon substitution of Equation 12.9 into Equation 12.8, the photon intensity is
𝐼=
𝑘2 𝑉̇ f [NO]sample
𝑘2 + 𝑘3 [M] + 𝑉̇ f
(12.10)
If the reactor is operated such that the ozone flow rate is large compared with the
sample flow rate, then
𝑉̇ f
= 𝑉̇ f,O3
[M] = [O3 ] = 𝑃∕𝑅𝑇
(12.11)
Fixing the reactor temperature fixes the chemical rate constants 𝑘1 , 𝑘2 , and 𝑘3 , and
fixing the pressure fixes the ozone concentration. Therefore, as indicated by Equation 12.10,
for a given volumetric flow rate of ozone and exhaust gas sample, the photon intensity is
proportional to the concentration of nitric oxide in the entering sample stream.
Exhaust Gas Oxygen Concentration
The exhaust gas oxygen concentration is measured with an oxygen sensor. The oxygen
sensor is used to control the air--fuel ratio, since the operation of a three-way catalyst
requires that the air--fuel ratio be maintained within about 1% of stoichiometric. The
sensor is constructed out of a thimble-shaped solid zirconium oxide (ZrO3 ) electrolyte
stabilized with yttrium oxide (Y2 O3 ). The development of the zirconia sensor is detailed
in Hamann et al. (1977). The interior and exterior surfaces of the electrolyte are coated
with porous platinum to form interior and exterior electrodes. Electrochemical reactions
on the electrodes produce negatively charged oxygen ions that then produce a voltage
across the electrolyte. The voltage 𝑉 produced depends on the oxygen ion flow rate that
in turn is proportional to the oxygen partial pressure at the electrodes, as indicated by
Equation 12.12. The symbol 𝐹 is the Faraday constant equal to 9.649 × 107 C/kmol. The
Exhaust Gas Analysis
363
0.8
1.4
Sensor voltage (mV)
1000
800
600
400
200
Figure 12.20 Oxygen sensor voltage versus
equivalence ratio.
0.6
oxygen partial pressure for a lean 𝜙 = 0.82 mixture is about 0.04 bar.
)
(
𝑃O2 atm
𝑅𝑇
𝑉 =
ln
4𝐹
𝑃O2 exhaust
1.0
1.2
(12.12)
The voltage output is highly nonlinear at stoichiometric conditions, with a large change
in the voltage between rich and lean conditions. If the exhaust mixture is rich, with a lack of
O2 , oxygen ions will flow from the air-side electrode across the electrolyte to the exhaust
side. If the mixture is lean, with excess oxygen, oxygen ions also form at the exhaust gas
electrode, and the migration of oxygen ions across the electrode drops. For rich conditions,
a voltage of about 800 mV is formed and for lean conditions about 50 mV is produced, as
indicated in Figure 12.20. A control set point voltage of about 0.5 V is used to maintain
stoichiometric conditions. If the sensor voltage is below the set point voltage, the exhaust
is considered by the control system to be lean, and vice versa. The behavior of the oxygen
sensor is temperature dependent, as the electrolyte needs to be above 280◦ C for proper
operation. A heating electrode is sometimes embedded in the sensor to rapidly bring it up
to operating temperature.
Particulates
There are a number of techniques used to characterize and measure particulate emissions.
These include light absorption, filter discoloration, and measurement of the total mass of
particulates trapped on a filter paper. The exhaust particle size distribution can be measured
using aerosol instruments such as the scanning mobility particle sizer (Wang and Flagan,
1990).
The absorption-type smoke meter uses the principle of light absorption by particles.
A pump is used to draw undiluted exhaust gas into a measuring chamber that has a light
source at one end and a photodiode at the opposite end. The attenuation of the beam of light
by the exhaust is proportional to the particle concentration. The filter-type smoke meter
draws a metered amount of exhaust gases through a filter paper. The blackening of the filter
paper is compared against a ‘‘Bacharach grey scale.’’
Standards, such as SAE J 1280, that employ direct mass measurement also specify
the use of a dilution tunnel in order to simulate the exhaust conditions near a vehicle. The
particulates leaving the exhaust pipe are at a relatively high-temperature and concentration
in the outlet exhaust flow. These gases cool during the mixing process with the atmosphere,
and the associated condensation and agglomeration processes will change the structure and
364
Engine Testing and Control
Engine
Exhaust
stream
Air
Figure 12.21 Exhaust
gas dilution tunnel.
Mixing zone
Sampling
zone
Blower
density of the particulates in the exhaust gases. Dilution tunnels are used to standardize this
near-field (< 3m) mixing process. A dilution tunnel is shown in Figure 12.21. The tunnel
is about 0.3 m in diameter. By flowing dilution air at a constant speed, typically 10 m/s,
through a converging--diverging nozzle, the Venturi effect can be used to remove exhaust
gas from the exhaust pipe. Minidilution tunnels with a 25-mm diameter have also been
developed. Downstream of the nozzle, the exhaust is well mixed with the dilution air. The
diluted exhaust gas is sampled and drawn through Teflon-coated glass fiber paper filters.
The total particulate mass is trapped by the filter found by the increase in weight of the
sample filter. In order to compute the dilution ratio, which is defined as the ratio of dilute
mixture flow rate to exhaust gas flow rate, the carbon dioxide concentration is measured in
both the engine exhaust and the diluted sample. The dilution ratio is typically about 10 : 1.
Fuel--Air Equivalence Ratio
To solve for the fuel--air equivalence ratio from exhaust gas analysis, let us write the
combustion reaction as
𝑎
C𝛼 H𝛽 O𝛾 N𝛿 + s (O2 + 3.76N2 ) →
𝜙
(12.13)
𝑛1 CO2 + 𝑛2 H2 O + 𝑛3 N2 + 𝑛4 O2 + 𝑛5 CO + 𝑛6 H2 + 𝑛7 CH𝑧 (g)
where the subscript 𝑧 is defined as
𝑧 = 𝛽∕𝛼
(12.14)
In Equation 12.13, CH𝑧 (g) represents the gaseous hydrocarbons that a flame ionization
detector records. The parameter 𝑧 is the average hydrogen to carbon ratio of the hydrocarbons and is unknown. In engines that function properly, the exhaust gas contains negligible
hydrocarbons, as far as atom balancing is concerned. They are included in the analysis
because they are important in engines that misfire. A carbon and oxygen balance on this
equation leads to Equation 12.15 for the equivalence ratio:
𝜙=
2 (1 + (1∕4)𝛽∕𝛼 − (1∕2)𝛾∕𝛼) (𝑦1 + 𝑦5 + 𝑦7 )
2𝑦1 + 𝑦2 + 2𝑦4 + 𝑦5
(12.15)
Notice that if we had an instrument to measure the mole fraction of water (𝑦2 ) in the exhaust
gas (generally we do not), then the equivalence ratio could be determined with no further
analysis or approximations. A complication arises in that most emission instruments do not
function if water condenses in them. A common way to handle this is to condense the water
from the sample prior to delivery to the instruments. With no water vapor in the sample, we
then say the concentration is ‘‘dry,’’ as opposed to ‘‘wet.’’ The dry concentration depends
Exhaust Gas Analysis
365
on the amount of water condensed out. Denoting a dry concentration with a superscript
zero, we can relate the dry concentration of any species 𝑖 to the wet concentration by
𝑦𝑖
(12.16)
𝑦0𝑖 =
1 − 𝑦2
Of course there is no physical significance to ‘‘dry water’’ (𝑖 = 2). It is mathematically
convenient, however, to speak of dry water defined by Equation 12.16. In terms of dry
concentrations, the equivalence ratio is
𝜙=
2 (1 + (1∕4)(𝛽∕𝛼) − (1∕2)(𝛾∕𝛼)) (𝑦01 + 𝑦05 + 𝑦07 )
2𝑦01 + 𝑦02 + 2𝑦04 + 𝑦05
(12.17)
There is no change in the function since the wet water term (1 − 𝑦2 ) factors out.
The water concentration is found from a hydrogen atom balance. However, that does
not solve the problem completely because it introduces two new unknowns: 𝑦6 , the hydrogen concentration; and 𝑧, the hydrogen to carbon ratio of the exhaust hydrocarbons,
Equation 12.15.
Spindt (1965) has found by experiments that it is satisfactory to assume
𝑦2 𝑦5
= 3.5
(12.18)
𝑦1 𝑦6
Substitution of these relationships into the hydrogen balance gives, after much manipulation
𝑦02 =
(1∕2)(𝛽∕𝛼)(𝑦01 + 𝑦05 )
1 + (𝑦05 ∕3.5 𝑦01 )
(12.19)
In experiments, the measured fuel--air ratios and those determined from exhaust gas analysis
agree to within ±2%. For greater accuracy, Lynch et al. (1997) recommend that the NO
concentration be measured and also included in the analysis.
Residual Fraction
The residual fraction can be determined directly by use of a sampling valve to withdraw
gases from the compression stroke for analysis with the same instruments already described.
The mole fraction of carbon dioxide in those gases is
𝑦CO2 = 𝑦r 𝑦′′
CO
2
(12.20)
is the carbon dioxide mole fraction in the
where 𝑦r is the residual mole fraction and 𝑦′′
CO2
exhaust gases. The residual mass fraction is, by Equation 12.20,
)]−1
( ′′
[
𝑦CO
𝑚′
2
−1
(12.21)
𝑓 = 1 + ′′
𝑚
𝑦CO2
The molecular weights of the residual gas M′′ and fuel--air mixture M′ are known if the
mole fractions of their constituent gases are known.
A typical sampling valve is shown in Figure 12.22. The seat (2) has threads that are
screwed into a receiving hole that provides access to the combustion chamber. In this
application, the valve is mounted in the cylinder head, and when it is opened, gases in the
cylinder are withdrawn. The poppet value (1) is opened by a trigger signal corresponding
to a given crank angle and is open for 1--2 ms. It is electromagnetically opened by passing
2--5 A of current through a coil (19). A spring (10) closes the valve when the current stops.
366
Engine Testing and Control
10
8 9
7
4
6
2
2’
11 13
12
3
5
21
23
20
1
15
Figure 12.22 Typical
sampling valve. (Courtesy
Tsukasa Sokkler, Ltd.)
22
14
17 18
16
24
19
27
26
28
29
30
25
A capacitance type of valve motion detector (24 and 25) is incorporated into the valve,
and the sampled gases flow out the port (4). By opening the valve at the same angle in
successive cycles, it is possible to have a steady flow of gas for delivery to the exhaust gas
analyzers.
It is also possible, though certainly more difficult, to determine the residual fraction
by measuring the temperature at some angle during the compression stroke and applying
the equation of state.
𝑃 𝑉 = 𝑚𝑅𝑇
(12.22)
For further information about engine testing and measurement, the reader is referred
to Plint and Martyr (2012).
12.5 CONTROL SYSTEMS IN ENGINES
The control systems in internal combustion engines operate using sensors, microprocessors,
and actuators. The sensors measure temperatures, pressures, flow rates, and concentrations
at various locations throughout the engine. The sensor information is provided to the engine
control microprocessors to characterize the instantaneous state of the engine. The control
systems are quite complex, as they interact with each other, are nonlinear, and need to have
robust operation over a wide range of transient speed and load conditions.
The parameters that characterize the engine state include, at a minimum, the engine
speed, throttle position, crank angle, intake airflow rate, intake manifold and ambient
pressure, inlet air and coolant temperature, exhaust gas oxygen concentration, and onset of
knock. The engine components that need to be controlled include not only the fuel injectors
or carburetor but also the spark plugs, the exhaust gas recirculation valve, the turbocharger
waste gate or variable area nozzle, and for a vehicle, the transmission. An example engine
control system diagram is presented in Figure 12.23.
There are two types of control systems used on engines: memory-based systems and
adaptive systems. Memory systems store the optimum values of control variables such as
spark timing and fuel injector pulse width for a range of engine operating conditions in a
table or map. The optimum values include both efficiency and emissions considerations.
For a given engine load (that is, manifold air pressure) and engine speed, the engine control
Control Systems in Engines
Dyno
control
board
Ambient temp
Load
bank
(+)
(–)
Torque
Dyno temp
Inlet air temp across engine
Armature current
Outlet air temp across engine
Fuel flow
Engine
control
Throttle actuator
Throttle position sensor
Intake manifold temp
Intake manifold pressure
Engine start
Digital signal
Engine stop
Digital signal
Cylinder pressure
Oil pressure
Crank encoder
Analog control signal
IGN timing control
Analog control signal
Fuel control
Engine
Emission
analysis
Gas sample
Intake
Airflow
Field current
Ambient BAP
Ambient R.H.
Dyno control voltage
367
Dyno
RPM signal
Exhaust
Coolant flow
Coolant temp inlet
Coolant temp outlet
Air flow across engine
Exhaust temps (x3)
A/F
meter
Air–fuel ratio
Exhaust gas information
Figure 12.23 Example diagram of engine control system. Adapted from Kirkpatrick and
Willson (1998).
computer will ‘‘look up’’ the optimum timing and then change the spark timing to that
optimum value. In feed-forward operation, the spark advance is computed from a spark
advance map as a function of engine speed and load. The spark advance map is determined
from engine calibration testing. Memory-based control systems have the disadvantage of
not accounting for part-to-part variation in engine components, the effect of deposits, and
fuel property changes. Also, the optima determined from mapping measurements on a test
engine are not exactly the same from engine to engine.
The engine control systems operate with both open- and closed-loop feedback control.
Some of the critical sensors, such as the oxygen sensor, operate properly only when an
engine has warmed up. When an engine is cold, it operates on an open-loop control without
input from the oxygen sensor. When the engine has warmed up, it switches to closed-loop
feedback control and uses the oxygen sensor data to compute the required fuel flow rate.
If the oxygen sensor indicates a rich mixture (𝜙 > 1), the pulse width of the fuel injector
actuator signal is reduced to decrease the equivalence ratio. After a time lag, primarily the
time required for the leaner fuel--air mixture to flow from the injector to the oxygen sensor
located in the exhaust system, the oxygen sensor will indicate a lean mixture (𝜙 < 1). In
response, the controller will increase the injection pulse width to enrich the mixture. With
this type of closed-loop control, known as a limit cycle, the air--fuel mixture continually
oscillates about stoichiometric conditions, with a time average value of 𝜙 < 1.
Adaptive systems determine an operating point from real-time measurement of engine
variables and subsequent correction of the look-up tables. The subsystems that have used
368
Engine Testing and Control
adaptive control are the exhaust gas recirculation, evaporative emissions, idle air control,
and air--fuel ratio control. Various calibrations need to be periodically reset due to wear,
aging, and replacement of components, such as fuel injectors or sensors.
Subsystems that alternate between open- and closed-loop control, such as the air--fuel
ratio, will perform periodic adaptive corrections. If a system is in closed-loop control, it
can compare the closed-loop values with the open-loop values. If there is a significant
change, the open-loop table values are corrected. The corrections to the look-up tables are
also obtained by driving the vehicle through specified driving cycles, typically stop-and-go
traffic with intermittent idle periods. During the driving cycle, the control system applies
very small perturbations to the parameters, such as the ignition timing, and measures the
response in other parameters, such as the fuel flow. Optimum values are then stored in the
various tables such as the ignition-timing table.
The U.S. Environmental Protection Agency has mandated the use of on-board diagnostics (OBD) on passenger cars and light- and heavy-duty trucks built after 1994. These
regulations are designed to detect emissions-related malfunctions. The diagnostic system
uses various methods to communicate diagnostic information to operators. If a fault is
detected, such as a faulty sensor, a fault code corresponding to the fault is stored. One type
of diagnostic system flashes the ‘‘check engine’’ light using a variation of the Morse code
signaling system. Alternatively, a computer or some similar type of digital analysis tool
can be connected to the communications port of the diagnostic system.
If a sensor fails, the engine control system is able to maintain engine operation. It
substitutes a fixed value for the sensor input, sends out a fault code, and continues to
monitor the incorrect sensor input. If the sensor returns to normal limits, the engine control
system will then return to processing the sensor data.
Some automotive engine control systems disable a number of fuel injectors if a cylinder
head sensor indicates that the engine is overheating, perhaps from a loss of coolant. Varying
and alternating the number of disabled fuel injectors controls the engine temperature. When
a fuel injector is disabled, its cylinder works as a convective heat exchanger since airflow
into and out of the cylinder continues to take place, and no combustion is occurring in the
cylinder. One consequence of this strategy is that the engine will produce proportionally
less power with disabled fuel injectors. Fuel injector disabling is also used if an engine or
vehicle over-speed condition is detected. Once the speed is reduced, the engine returns to
normal operating mode.
The onset of knock is detected by a knock detector. The methods used to detect knock
include piezoelectric and magnetostrictive techniques. If a knock signal is sent to the engine
control unit, the timing will be retarded, and then the throttle will be closed until the knock
ceases. The delay in ignition will reduce the torque produced by the engine.
The sensor information stored by the engine control unit can also be sent via telemetry or
wireless Internet to a host computer. Racing teams use this technique to debug and fine-tune
high-performance race cars. Many rental car and trucking firms are tracking the operation
of their vehicles in this manner. In the near future, wireless communication technologies
will be used by engine and vehicle manufacturers not only for routine engine diagnostics
but also to collect information about long-term engine performance and reliability.
Recent developments in control systems are the development of parametric engine
performance models, two examples of which are mean value models and discrete-time
models. The mean value engine model (MVEM) is physics based, and is an intermediate
level engine model that has been used in applications where overall engine parameters such
as engine efficiency, emissions, airflow rate, air--fuel ratio, maximum cylinder pressure,
and exhaust pressure are of primary interest, as opposed to crank angle resolved behavior.
The MVEM models are highly compatible with model-based engine control systems, as
Vehicle Emissions Testing
369
they predict the cycle-averaged engine behavior. Calibration and validation of MVEMs is
accomplished both by comparison to more complex engine models and engine test data.
A classic paper outlining engine control systems is by Cook and Powell (1988). For
further information about recent developments in engine control systems, the reader is
referred to books by Guzzella and Onder (2010), and by Ulsoy et al. (2012).
12.6 VEHICLE EMISSIONS TESTING
Speed (mph)
For emissions testing of engines in vehicles, a chassis dynamometer is used. The chassis
dynamometer is used to put vehicles through a driving cycle, with the advantage of not
having to instrument a moving vehicle. Chassis dynamometers were first developed for locomotives and more recently for road vehicles. The U.S. Environmental Protection Agency
requires chassis dynamometer testing of many classes of vehicular engines for emissions
purposes. The word ‘‘homologation’’ is used to describe this certification process.
The chassis dynamometer is composed of a series of rollers, flywheels, and dynamometers. The vehicle to be tested is driven onto the top of the chassis dynamometer and its drive
tires rotate between two rollers that are mechanically connected to flywheels and electric
dynamometers. The rolling inertia of the vehicle is simulated with rotating flywheels and
electronic inertia. A cooling fan is used to produce adequate airflow to prevent the engine
from overheating.
The United States, the European Community, and Japan have developed their own
driving cycles that simulate a variety of driving conditions for various classes of vehicles.
Two United States driving schedules are shown in Figure 12.24. The United States driving
cycle for passenger cars and light-duty trucks is the Federal Test Procedure (FTP). The test
90
80
70
60
50
40
30
20
10
0
0
200
400
600
800
Time (s)
1000
1200
1400
Speed (mph)
(a)
Figure 12.24 (a) Federal
Test Procedure LA4 driving
schedule. (b) US06 driving
schedule.
90
80
70
60
50
40
30
20
10
0
0
100
200
300
Time (s)
(b)
400
500
600
370
Engine Testing and Control
procedure for heavy-duty (gross vehicle weight > 8500 lbs.) highway engines is the EPA
transient test procedure and the EPA smoke test procedure.
12.7 REFERENCES
BROWN, W. L. (1967), ‘‘Methods for Evaluating Requirements and Errors in Cylinder Pressure
Measurement,’’ SAE paper 670008.
CHEUNG, H. and J. HEYWOOD (1993), ‘‘Evaluation of a One-Zone Burn Rate Analysis Procedure Using
Production SI Engine Pressure Data,’’ SAE paper 932749.
COOK, J. and B. POWELL (1988), ‘‘Modeling of an Internal Combustion Engine for Control Analysis,’’
IEEE Control Syst. Mag., August, pp. 20--26.
FOSTER, D. (1985), ‘‘An Overview of Zero-Dimensional Thermodynamic Models for IC Engine Data
Analysis,’’ SAE Technical Paper 852070.
GUZZELLA, L. and C. ONDER (2010), Introduction to Modeling and Control of Internal Combustion
Engine Systems, Springer-Verlag, Berlin.
HAMANN, E., H. MANGER, and L. STEINKE (1977), ‘‘Lambda-sensor with Y2 O3 -Stabilized ZrO2 Ceramic for Application in Automotive Emission Control Systems,’’ SAE paper 770401.
KASTNER, L. J. (1947), ‘‘An Investigation of the Airbox Method of Measuring the Air Consumption
of Internal Combustion Engines,’’ Proc. Inst. Mech. Eng, Vol. 157, pp. 387--404.
KIRKPATRICK, A. and B. WILLSON (1998), ‘‘Computation and Experimentation on the Web with
Application to Internal Combustion Engines,’’ ASEE J. Eng. Educ., Vol. 87, No. 5, pp. 529-537.
LANCASTER, D. R., R. B. KRIEGER, and J. H. LIENESCH (1975), ‘‘Measurement and Analysis of Engine
Pressure Data,’’ SAE paper 750026.
LYNCH, D. and W. SMITH (1997), ‘‘Comparison of AFR Calculation Methods Using Gas Analysis and
Mass Flow Measurement,’’ SAE paper 971013.
PLINT, M. and A. MARTYR (2012), Engine Testing: Theory and Practice, 4th Edition, Elsevier, Ltd.,
Oxford.
RANDOLPH, A. (1990), ‘‘Methods of Processing Cylinder-Pressure Transducer Signals to Maximize
Data Accuracy,’’ SAE paper 900170.
SPINDT, R. S. (1965), ‘‘Air--Fuel Ratios from Exhaust Gas Analysis,’’ SAE paper 650507.
STONE, C. R. (1989), ‘‘Airflow Measurement in Internal Combustion Engines,’’ SAE paper 890242.
WANG, S. and R. FLAGAN (1990), ‘‘Scanning Electrical Mobility Spectrometer,’’ Aerosol Sci. Tech.,
Vol. 13, pp. 257--261.
ULSOY, A., H. PENG, and M. CAKMAKCI (2012), Automotive Control Systems, Cambridge University
Press, New York.
12.8 HOMEWORK
12.1
Measurements of the exhaust gases of a hydrogen-fueled engine indicate a composition of
71.0% N2 , 25.0% O2 , and 3.9% H2 O. At what equivalence ratio was the engine operated?
12.2
The exhaust composition of a test engine is as follows:
CO2 = 11.5%
H2 O = 7.11%
N2 = 77.99%
O2 = 3.19%
CO = 0.06%
H2 = 0.01%
NO = 310 ppm
NO2 = 20 ppm
CH4 = 350 ppm
C3 H8 = 225 ppm
C7 H17 = 475 ppm
Homework
371
Find the following:
(a) Wet concentration in ppm of HC and NO𝑥 as would be indicated by heated
flame ionization and chemiluminescence detectors, respectively. (Assume the
FID responds to all carbon atoms equally.)
(b) Dry concentrations of CO2 , O2 in percent, and CO in ppm.
12.3
(c) Fuel--air equivalence ratio if the hydrogen to carbon ratio of the fuel is 1.3.
Explain how Equation 12.22 could be used to measure the residual fraction.
12.4
A diesel engine operated on C14 H27 produced exhaust gas of the following dry composition:
CO2 = 6.22%
O2 = 12.20%
CO = 0.024%
12.5
N2 = 81.51%
NO𝑥 = 400 ppm
HC = 200 ppm C
(a) Explain how the method of hydrocarbon measurement can yield a situation
wherein the sum of the exhaust constituents adds up to slightly greater than
100%.
(b) At what equivalence ratio was the engine operated? How would the answer differ
if one neglected the carbon monoxide and hydrocarbons?
An isooctane-fueled engine has a measured fuel mass flow rate of 0.5 g/s and air mass flow
rate of 7.0 g/s. The exhaust gas composition (dry) is measured to be CO2 = 11 % and CO =
3.0%. Compare the equivalence ratio computed from the exhaust gas composition with that
from the fuel--airflow rate ratio. Assume an equilibrium exhaust composition to estimate
the exhaust H2 concentration.
12.6
A test engine operates on methane at a mass flow rate of 2.0 g/s with an equivalence ratio
𝜙 = 0.8. (a) What is the inlet air mass flow rate? (b) If the exhaust is at standard conditions,
what are the volumetric flow rates of the exhaust products N2 , H2 O, CO2 , and O2 ?
12.7
Manufacturers of laminar airflow meters typically provide a calibration curve of the following form:
𝑉̇ stp = 𝑐1 Δ𝑃 + 𝑐2 Δ𝑃 2
where 𝑉̇ stp is the volumetric flow rate at standard temperature (298.15 K), and pressure
(1 bar), and Δ𝑃 is the pressure drop across the meter.
(a) Use dimensional analysis to show how the constants 𝑐1 and 𝑐2 would change for
measurements made at conditions other than standard temperature and pressure.
12.8
12.9
(b) How can one determine the mass flow rate rather than the volumetric flow rate?
Assuming one-dimensional, isentropic steady flow of an ideal gas with constant specific
heats, derive an expression for the constant 𝐶 of the critical flow nozzle in Figure 12.6.
The calibration constant depends on the nozzle throat area 𝐴, the gas constant 𝑅, and the
ratio of specific heats 𝛾. You may assume the upstream area is large enough that measured
𝑃1 and 𝑇1 are stagnation properties.
Figure 12.12 is a plot of log 𝑃 versus log V. Estimate the polytropic exponents in the
expression 𝑃 𝑉 𝑛 = constant in the middle of both the expansion and compression strokes.
How do these exponents relate to the specific heat ratio?
Chapter
13
Overall Engine Performance
13.1 INTRODUCTION
In this chapter, we take an overall view of the performance of internal combustion engines.
We use the information about friction, heat transfer, and combustion presented in the
previous chapters to explain and discuss the influence of various factors, such as engine
and piston speed, load, compression ratio, and ignition timing. Performance maps for
various representative spark ignition and compression ignition engines are introduced. The
frictional and aerodynamic drag components of road load are also discussed for application
to vehicle performance simulation.
13.2 EFFECT OF ENGINE AND PISTON SPEED
The effects of engine speed on the power and coolant load of an automotive spark ignition
engine at full load, that is, wide open throttle, are shown in Figure 13.1. The graphs in
Figure 13.1 plot the performance of an unthrottled 4.7 L V-8 spark ignition engine with
a bore 𝑏 = 95.2 mm and stroke 𝑠 = 86 mm at three different compression ratios, 𝑟 = 8,
10, and 12. Note that the indicated specific fuel consumption decreases with increasing
engine speed and then levels out. This is mainly because the percentage heat loss to the
coolant is decreasing with increasing engine speed. On the other hand, the brake specific
fuel consumption (bsfc) is fairly flat or decreasing at low speeds and is increasing at higher
speeds. At higher speeds, the friction and pumping losses become more significant, as they
increase quadratically with engine speed.
The power curves in Figure 13.1 are conveniently explained in terms of the expression
relating the power 𝑊̇ b of an engine to the volumetric efficiency 𝑒v , the net indicated thermal
efficiency 𝜂i , and the mechanical efficiency 𝜂mech . The relationship (four stroke) is given
in Equation 13.1:
𝑊̇ b = 𝜂mech 𝑊̇ i
(13.1)
FA
𝑒v 𝜌i 𝑉d 𝑁∕2
1 + FA
where, as before, FA is the fuel--air ratio, 𝑎o is the maximum available energy of combustion,
and 𝜌i is the mixture density in the intake manifold.
Figure 13.1 also gives the indicated power as a function of engine speed. Since the
engine is unthrottled, we can assume for qualitative purposes that the curve is the net
indicated power versus engine speed. All the terms multiplying the mechanical efficiency
= 𝜂mech 𝜂i 𝑎o
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
372
Effect of Air--Fuel Ratio and Load
373
r = 12
100
60
40
400
r=8
350
30
20
10
1000
10
300
12
10
120
8
100
.
Wi
80
300
60
40
r=8
275
10
250
12
250
2000
3000
Engine speed (rpm)
4000
1000
2000
3000
Engine speed (rpm)
225
isfc (g/kWh)
70
50
r = 12
140
8
bsfc (g/kWh)
Brake power (kW)
80
10
.
Wb
Indicated power (kW)
90
4000
Figure 13.1 Performance of a V-8 spark ignition engine at three different compression ratios.
(Adapted from Roensch, 1949.)
in Equation 13.1 constitute the net indicated power. If the indicated torque were constant,
then the indicated power would increase linearly with engine speed.
Torque as a function of engine speed usually reflects the variation in volumetric
efficiency with engine speed. A falling off in volumetric efficiency at the higher speeds
causes the falling off of the indicated power at high engine speeds. Recall that the speed at
which the volumetric efficiency peaks is dependent upon the valve timing. The indicated
power is not decreasing as fast as it would if the volumetric efficiency were the only
parameter changing with speed. The indicated efficiency is increasing slightly with speed.
The brake power is the product of the net indicated power and the mechanical efficiency. Friction power increases with the square of engine speed, since the friction torque
(proportional to brake mean effective pressure (bmep)) increases linearly with engine speed.
The mechanical efficiency therefore decreases linearly with engine speed, causing the brake
power to exhibit a maximum even though the indicated power is still increasing.
It was stated earlier that these generalizations are expected to apply to all engines. That
statement needs qualification in the case of two-stroke engines, especially carbureted ones.
Recall that in two-stroke engines, there is a significant difference between the delivered
mass and the trapped mass, because of short-circuiting. Any fuel that is short-circuited is
wasted and represents a loss not discussed in the context of Figure 13.1, since this effect
is usually negligible in four-stroke engines. As the trapping efficiency generally increases
with engine speed, the amount of fuel short-circuited will decrease with engine speed.
13.3 EFFECT OF AIR--FUEL RATIO AND LOAD
The effect of the air--fuel ratio on brake specific fuel consumption of a spark ignition
engine at a constant load is shown in Figure 13.2 for three different compression ratios,
(𝑟 = 8, 9, and 11). Figure 13.2 indicates that the spark ignition engine is most efficient when
running stoichiometrically or slightly lean. At very lean fuel--air ratios, the engine wastes
fuel because of misfire, and at rich fuel--air ratios it wastes fuel since there is not enough
oxygen present to liberate all of the fuels energy.
The effect of fuel--air ratio on the brake specific fuel consumption and exhaust emissions of a number of indirect-injection (IDI) and direct-injection (DI) compression ignition
Overall Engine Performance
bsfc (g/kWh)
480
Compression
ratio r
460
8
440
9
420
400
11
380
Figure 13.2 Brake specific fuel
consumption of a single cylinder
research engine versus air--fuel
ratio. (Adapted from French, 1983.)
Rich
11
12
13
Lean
14
15
16
Air–fuel ratio
17
18
6
5
4
3
2
1
0
380
340
300
bsfc
280
1500
HC
1000
50
HC (ppm)
100
0
500
NOx
0
CO
0.2
0.1
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07
Fuel–air ratio
0
CO (%)
0.3
Figure 13.3 The effect of
air--fuel ratio on the bsfc
and exhaust emissions of a
number of IDI and DI diesel
engines. (Adapted from
Motoyoshi et al., 1976.)
bsfc (g/kWh)
Smoke
Smoke (BSN)
engines is plotted in Figure 13.3. Since the stoichiometric fuel--air ratio for diesel fuel is
about 0.07, the equivalence ratio for the data plotted ranges from 0.14 to 0.85. As the mixture is leaned out, the NO𝑥 decrease, due to the lower combustion temperatures. The smoke
readings are in Bosch smoke number (BSN) units, a scale that measures the reflectivity of
a piece of filter paper through which some of the exhaust gas is passed.
NOx (ppm)
374
Effect of Air--Fuel Ratio and Load
375
1.80 L gasoline
b = 88.9 mm
s = 60.3 mm
2.21 L diesel
b = s = 88.9 mm
487
Brake specific fuel consumption (g/kWh)
426
Figure 13.4 Comparison of an
SI engine with an IDI--CI engine
design to produce equal torque
speed characteristics. (Adapted
from Walder, 1965.)
365
304
4000 rpm
243
487
426
365
2000 rpm
304
243
0
2
4
6
8 10 12 14
Brake mean effective pressure (bar)
The effect of load, that is, the brake mean effective pressure, on the brake specific
fuel consumption is qualitatively the same for both compression ignition and spark ignition
engines, as shown by representative examples in Figure 13.4, which compares a gasoline
and a diesel engine, and Figure 13.5 for a marine diesel with two types of prechambers. In
both cases, the bsfc will be infinite at idle, since the engine is producing no useful work but
is consuming fuel. As the load increases, the brake specific fuel consumption drops, goes
through a minimum, and may or may not increase depending on how the load is increased
at this point.
In the case of spark ignition engines, opening the throttle and increasing the delivery
ratio increases the load. This has little effect on the indicated efficiency, slightly increases
the friction, and significantly reduces the pumping losses. Again, the dominant factor is
the increase in mechanical efficiency. At constant fuel--air ratio, the brake specific fuel
consumption drops with increasing load all the way to the point of maximum load so
long as the imep increases faster than the fmep. In engines running at a fuel--air ratio less
than that corresponding to maximum power (about 𝜙 = 1.1, as we saw in our studies with
fuel--air cycles), the load can be increased further by increasing the fuel--air ratio. This
causes the brake specific fuel consumption to begin increasing with load once the engine
is running rich.
In the case of compression ignition engines, increasing the fuel--air ratio increases
the load; although this slightly drops the indicated efficiency and slightly increases the
376
Overall Engine Performance
Conventional prechamber
Variable geometry prechamber
130
272
110
258
90
245
70
231
50
217
30
4
5
8
10
12
bmep (bar)
14
Peak combustion pressure (bar)
Specific fuel consumption (g/kWh)
bar
305
16
Figure 13.5 Brake specific fuel consumption of a marine diesel engine versus load (Hermann,
1980).
friction, the increase in mechanical efficiency is so great that it improves the specific fuel
consumption. Just before the load is about to become smoke limited, the brake specific
fuel consumption begins to increase because significant quantities of fuel begin to be only
partially oxidized and thus are wasted.
The variable geometry prechamber of Figure 13.5 improved the fuel economy of the
marine engine by about 10 g/kWh or about 5%. The 5% improvement is small and does not
affect the trend shown for brake specific fuel consumption with load. Nevertheless, from an
economic perspective that 5% improvement is very significant since the cost of fuel saved
by a ship on a trip, although small compared with the total fuel cost, will be comparable to
the trip’s profit.
13.4 ENGINE PERFORMANCE MAPS
A common way of presenting the effects of both speed and load on engine performance
is shown in Figure 13.6, an engine performance map. The engine speed 𝑁 or the mean
piston speed 𝑈̄ p is plotted on the 𝑥-axis, and the brake mean effective pressure is plotted
377
Engine Performance Maps
12
bsfc (g/kWh)
bmep (bar)
10
280
8
300
335
6
400
4
Figure 13.6 Performance map of
bmep and bsfc versus mean piston
speed for an automotive spark
ignition engine.
500
2
750
2
4
6
8
10
12 14
Piston speed (m/s)
16
18
on the 𝑦-axis. Contour lines of constant brake specific fuel consumption are plotted on this
load--speed plane. The lines of constant bsfc are usually size independent for a given engine
family, so performance maps can be used to match an engine with a given load profile. The
upper envelope on a map is the wide open throttle performance curve. Its shape reflects
variations in the volumetric efficiency with engine speed, although small changes in inlet
air density are also involved. Performance maps can also be generated for emissions levels,
with contour lines of constant emission level.
A representative spark ignition automotive engine performance map is given in Figure 13.6. The engine was a turbocharged four-cylinder engine with a 92-mm bore and
80-mm stroke, and compression ratio of 𝑟 = 8.7. For general automotive applications,
engines are designed to have the region of minimum bsfc located at relatively low engine speeds (40--60% of maximum engine speed) and at relatively high loads (60--80% of
maximum bmep).
Representative diesel performance maps are given in Figure 13.7 as a function of engine
speed, and in Figure 13.8 as a function of piston speed. With a diesel engine, the power is
smoke limited, as the fuel--air ratio needs to be reduced relative to stoichiometric to reduce
exhaust particulates. The relative position of the point of minimum fuel consumption can
be moved up or down depending on the degree of mixture enrichment allowed.
At lower loads, the diesel bsfc is less than the SI engine bsfc due to its lower pumping
work and leaner air--fuel ratio. Increasing the fuel--air ratio increases the load; although
this slightly drops the indicated efficiency and slightly increases the friction, the increase in
mechanical efficiency is so great that it improves the specific fuel consumption. Just before
the load is about to become smoke limited, the brake specific fuel consumption begins to
increase because significant quantities of fuel begin to be only partially oxidized and thus
are wasted.
Engine performance maps generally have a single-valued minimum brake specific
fuel consumption operating point. Starting at the location of minimum bsfc on the map,
the fuel consumption increases in all directions. If one increases the engine speed, the
fuel consumption increases because of an increase in the friction loss. If one decreases the
engine speed, the fuel consumption increases because of an increase in the heat loss per
cycle. At very lean fuel--air ratios, the engine wastes fuel because of misfire, and at rich
fuel--air ratios it wastes fuel since there is not enough oxygen present to liberate all of the
fuel’s energy. If one increases the load, the fuel consumption increases because the mixture
must be enriched beyond stoichiometric. If one decreases the load, the fuel consumption
Overall Engine Performance
Figure 13.7 Performance map of
bmep and bsfc versus engine speed
for a representative automobile
diesel engine.
9
bsfc (g/kWh)
8
7
250
6
bmep (bar)
378
275
325
5
300
4
350
400
3
500
2
750
1
0
2
4
6
8
10
Piston speed (m/s)
12
14
16
Figure 13.8 Performance map of a four-cylinder naturally aspirated indirect-injection (NA-IDI)
diesel engine: 𝑏 = 76.5 mm, 𝑠 = 80 mm and 𝑟 = 23. (Adapted from Hofbauer and Sator, 1977.)
increases because the friction is becoming a larger proportion of the indicated work. Finally,
for a spark ignition engine with an intake throttle valve, as the load decreases, the throttle
is closed, increasing the throttling losses, and increasing the bsfc.
Performance maps for large stationary and marine engines are not typically produced,
since such engines usually operate at one speed. For more information about the use of
automotive class spark ignition performance maps, the reader is referred to Shayler et al.
(1999).
Effect of Engine Size
379
13.5 EFFECT OF ENGINE SIZE
We now look at the effect of engine size. The torque an engine will produce, by definition
of the mean effective pressure, is
𝜏b =
1
bmep 𝑉d
4𝜋
(four-stroke engine)
(13.2)
𝜏b =
1
bmep 𝑉d
2𝜋
(two-stroke engine)
(13.3)
The power can also be expressed in terms of the mean effective pressure:
1
𝑊̇ b = bmep 𝐴p 𝑈̄ p
4
(four-stroke engine)
(13.4)
1
𝑊̇ b = bmep 𝐴p 𝑈̄ p
2
(two-stroke engine)
(13.5)
Therefore, for a given stress level (bmep, 𝑈̄ p ), the torque is proportional to the displacement
volume 𝑉d and the power is proportional to the piston area 𝐴p
𝜋
𝑉d = 𝑛c 𝑏2 𝑠
4
𝜋
𝐴p = 𝑛c 𝑏2
4
(13.6)
(13.7)
Finally, we can also write for four- and two-stroke engines:
𝑚̇ f = bsf c ⋅ bmep ⋅ 𝑉d ⋅ 𝑁∕2
𝑚̇ f = bsf c ⋅ bmep ⋅ 𝑉d ⋅ 𝑁
bsf c =
(four-stroke engine)
(two-stroke engine)
𝑒v 𝜌 i
FA
1 + FA bmep
(13.8)
(13.9)
(13.10)
where, as before, FA is the fuel--air ratio, 𝑒v is the volumetric efficiency, and 𝜌i is the
mixture density in the intake manifold. Notice that Equation 13.10 does not explicitly
include engine size, so the efficiency of engines is expected to be a weak function of size
for a given stress.
The specific fuel consumption versus cylinder bore for representative diesel engines
is shown in Figure 13.9. This figure is based on two- and four-stroke designs with bores
from 62 to 900 mm. For bores greater than 500 mm, the thermal efficiency is about 50%.
The ratio of the maximum bore to minimum bore is about 15, corresponding to a 3400 to
1 displacement volume ratio, whereas the brake specific fuel consumption varies by only
a factor of 1.6. Although it is indeed a weak function with respect to the bore, the change
in specific fuel consumption with the bore is significant with respect to fuel economy. An
important factor underlying the trend shown in Figure 13.9 is that the surface to volume
ratio of the cylinder is decreasing with increasing bore:
(𝐴∕𝑉 ) ∼ 𝑏−1
This means that there will be proportionally less heat lost as the bore increases. Another
factor working in the favor of large engines is that the rotational speed decreases with the
bore size for a constant piston speed,
𝑁 ∼ 𝑠−1 ∼ 𝑏−1
380
Overall Engine Performance
Figure 13.9 Brake specific
fuel consumption of two- and
four-stroke engines versus
cylinder bore. (Thomas et al.,
1984.)
Specific fuel consumption (g/kWh)
280
260
240
220
200
180
160
0
100
200
300 400 500 600
Cylinder bore (mm)
700
800
900
so that there is more time near top center for fuel injection and combustion. This means
that there will be less of a volume change during combustion and thus there will be a closer
approach to constant volume combustion.
As discussed in Chapter 11, measurements indicate that the motoring mean effective
pressure decreases as engine size increases, Equation 13.11:
mmep = 1.3 × 105
𝑈̄ p 𝜇
𝑏
+ 1.7 × 108 (𝑟 + 15)
𝜇2
𝜌𝑏2
(13.11)
Therefore, the friction can be expected to be relatively less in large engines than in small
engines. The brake thermal efficiency of the state-of-the-art large (𝑏 > 500 mm) diesel
engines is about 50%. Surely, they are among the most efficient engines in the world. Their
low losses due to heat transfer, combustion, and friction have already been mentioned. They
also use late-closing intake valves to realize a longer expansion stroke than compression
stroke.
The generalizations just drawn ought to hold true for spark ignition engines too,
although the point is academic, for they are not practical unless care is taken to stratify
the charge. Large homogeneous charge spark ignition engines are not practical because
their octane requirements are too high. In earlier chapters, it was pointed out that the flame
speed is in part controlled by the magnitude of the turbulence, and that the turbulence is
proportional to the piston speed. It follows that the combustion duration in crank angle is
constant, but in the time domain is inversely proportional to the engine’s rotational speed.
There is consequently more time for knock precursors to form in relatively low speed,
large engines. For similar reasons, the engineering of small high-speed diesel engines is a
challenge, as there is little time for autoignition to occur and/or inject the fuel at reasonable
pressures.
13.6 EFFECT OF IGNITION AND INJECTION TIMING
For spark ignition gasoline engines, the timing parameter is the spark timing, and for diesel
engines the timing parameter is the fuel injection timing. A classic plot of the effect of
spark timing on the brake mean effective pressure for a number of automotive engines at
different chassis dynamometer speeds is given in Figure 13.10. Note that the variations in
spark timing have the same percentage effect on the bmep at all the tested dynamometer
bmep scale – 1 division = 10%
Effect of Ignition and Injection Timing
–20
15 km/h
60 km/h
30 km/h
80 km/h
50 km/h
100 km/h
381
Spark timing for 98%
maximum power
–10
Retard
0
+10
+20
Advance
–20
–10
Retard
0
+10
+20
Advance
Figure 13.10 Effect of spark timing on bmep for a number of different chassis dynamometer
speeds. (Adapted from Barber, 1948.)
speeds. The data are well correlated by Equation 13.12:
bmep
= 1 − (Δ𝜃∕53)2
(bmep)max
(13.12)
where Δ𝜃 is the change in degrees of crank angle from the angle of maximum bmep.
Although the data correlated are rather old, they are still representative of today’s engines.
Engines today are usually timed to a crank angle referred to as MBT (minimum advance for
best torque). Examine Figure 13.10 and notice how relatively flat the bmep curve is in the
vicinity of the maximum. Now, reexamine Figure 7.30 and notice how sensitive the nitric
oxide emissions are to variations in spark timing. Clearly, if the timing is slightly retarded,
say 5◦ from that of maximum bmep, then the engine power will decrease very little, and
under some operating conditions the nitric oxides will be greatly reduced. Retarded timing
also somewhat reduces the engine’s octane requirement.
The term MBT spark timing is widely accepted, yet there is no quantitative definition in terms of how far the spark should be retarded from the point of maximum torque.
Therefore, we will define MBT timing as a spark retard of 4◦ from the angle of maximum torque. This definition agrees with values reported in the literature to a tolerance of
about ±2◦ .
Figure 13.11 shows how the MBT timing can be expected to vary with engine speed,
equivalence ratio, and residual mass fraction. Because the charge is diluted by either air
(in which case it is lean) or exhaust gas, the combustion duration and ignition delay both
increase, thereby requiring a greater spark advance to more or less center the combustion
about top center. Likewise, as engine speed increases, ignition delay and the MBT spark
advance increase.
One way to illustrate the trade-offs involved in controlling nitric oxides by retarding the
fuel injection timing is to plot the brake specific nitric oxides versus the brake specific fuel
consumption at full load as in Figure 13.12. The graph shows the response of two different
Overall Engine Performance
70
Figure 13.11 Minimum spark
advance for best torque.
(Adapted from Young, 1980.)
1.0
f = 0.40
60
50
0.8
40
f = 0.10
Equivalence ratio
Minimum spark advance for best torque (CA deg)
382
0.9
30
1.0
20
10
1000
1200
1400
1600
1800
Engine speed (rpm)
2000
2200
Figure 13.12 Brake specific nitric
oxide emissions versus brake
specific fuel consumption at full
load as fuel injection timing is varied.
(Adapted from Pischinger and
Cartelleri, 1972.)
diesel engines to changes in timing at full load and rated speed of 𝑁 = 2800 rpm. The
engines are production, in-line six-cylinder, naturally aspirated four-stroke diesel engines
with 𝑉d = 5.9 L and a compression ratio of 𝑟 = 17. For the direct-injection engine, a
significant reduction in the nitric oxide emissions can be realized at the expense of a slight
increase in the brake specific fuel consumption. With indirect-injection engines, there is
no appreciable change in the nitric oxides as injection timing is changed. However, at part
loads the IDI engine shows response curves more like those shown for the direct injection
engines. Retarding the injection timing is an effective means of controlling the nitric
oxide emissions, but with diesel engines, it is usually at the expense of an increase in the
particulate or smoke emissions. Furthermore, with indirect injection, retarding the timing
does not always reduce the nitric oxides. The results discussed point out that it is more
Vehicle Performance Simulation
383
Thermal efficiency (%)
58
Figure 13.13 Thermal efficiency of a spark
ignition engine as a function of the nominal
compression ratio. (Adapted from Caris and
Nelson, 1959.)
Otto fuel–air cycle
54
50
Indicated
46
42
Brake
38
34
0
10
12
14
16
Compression ratio
18
20
difficult to generalize about the performance of diesel engines than about gasoline engines.
This is because there are many more degrees of freedom available in the design of a diesel
engine.
13.7 EFFECT OF COMPRESSION RATIO
As shown previously in Figure 13.1, increasing the compression ratio decreases the brake
specific fuel consumption. The particular results shown are, of course, unique to the specific
engine design tested. The compression ratio trends depicted and their underlying causes,
however, are typical to all engines, compression or spark-ignited engines, two or four-stroke
engines.
Figure 13.13 indicates that the indicated specific fuel consumption improves at a
faster rate with increasing compression ratio than the brake specific fuel consumption,
because both friction and heat losses are increasing with compression ratio. In fact, there
is an optimum compression ratio due to these effects. The spark advance is set for best
efficiency, as is the fuel--air equivalence ratio at a lean setting of 𝜙 = 0.91. The compression
ratios of spark ignition engines are less than the optima shown in Figure 13.13 to avoid
knock.
Computer simulations of diesel engines show similar trends (McAulay et al., 1965).
An optimum compression ratio of 12--18 is typical, and is the underlying reason why
direct-injection diesel engines have compression ratios in the same range. The compression
ratios of indirect-injection diesel engines are greater than optimum to assist in cold starting,
which is harder than with direct-injection engines because of the high heat loss in the
prechamber.
13.8 VEHICLE PERFORMANCE SIMULATION
We finish this chapter with a vehicular application. For a vehicle, the power requirements that need to be met by an engine are specified by a road load power equation,
Equation 13.13, which includes the effects of aerodynamic drag and rolling resistance:
∑
𝑊̇ v = 𝐹 ⋅ 𝑈v
(13.13)
= (𝐶r 𝑚v 𝑔 + 21 𝐶d 𝜌o 𝐴v 𝑈v2 ) 𝑈v
384
Overall Engine Performance
where
𝐶r = coefficient of rolling resistance
𝑚v = mass of vehicle (kg)
𝑔 = gravitational constant, 9.81 m/s2
𝐶d = drag coefficient
𝐴v = vehicle front cross-sectional area, (m2 )
𝑈v = vehicle speed (m/s)
Automotive engines are expected to operate well over a wide range of speeds and loads.
Figure 12.24 in Chapter 12 presents two driving cycles used by the U.S. Environmental
Protection Agency for regulatory purposes. In each case, vehicle speed as a function of
time is specified. A vehicle simulation using Equation 13.13 can be used to assess the
fuel economy performance of various candidate engines and vehicle combinations. From
knowledge of the vehicle’s characteristics, including frontal area, drag coefficient, weight,
and gear ratios, the driving cycle can be transformed into a specification of the required
engine torque and speed as a function of time.
The total fuel consumed by the engine during the driving cycle will be the integral of
the fuel flow rate, Equation 13.14:
𝑚f =
∫0
𝑡
𝑚̇ f (𝑡) 𝑑𝑡 =
𝐴p
4 ∫0
𝑡
bsf c(𝑡) bmep(𝑡) 𝑈̄ p (𝑡) 𝑑𝑡
(13.14)
For a two-stroke engine, the factor of four would instead be a factor of two. In order
to do the integration, one needs bsfc, bmep, and 𝑈p as functions of time. The latter two are
known since the engine torque and speed is known from the driving cycle requirements
and the vehicle characteristics.
The brake specific fuel consumption can be determined for each load and speed point
of the driving cycle from the engine performance map. If an emissions map is available, a
similar computation can be performed to compute the total emissions produced during the
driving cycle. For further information about the trends in the performance characteristics
of modern automobile engines, the reader is referred to Heywood and Welling (2009).
13.9 REFERENCES
BARBER, E. M. (1948), ‘‘Knock Limited Performance of Several Automobile Engines,’’ SAE Trans.,
Vol. 2, p. 401.
CARIS, D. F. and E. E. NELSON (1959), ‘‘A New Look at High Compression Engines,’’ SAE Trans.,
Vol. 67, p. 112.
FRENCH, C. (1983), ‘‘A Universal Test Engine for Combustion Research,’’ SAE paper 830453.
HERMANN, R. (1980), ‘‘PA4-200 Engines with Variable Geometry Precombustion Chamber and Two
Stage Turbocharging System,’’ ASME paper 80-DGP-22.
HEYWOOD, J. and O. WELLING (2009), ‘‘Trends in Performance Characteristics of Modern Automobile
SI and Diesel Engines,’’ SAE paper 2009-01-1892.
HOFBAUER, P. and K. SATOR (1977), ‘‘Advanced Automotive Power Systems, Part 2: A Diesel for a
Subcompact Car,’’ SAE Trans., Vol. 86, paper 770113.
MCAULAY, K., T. WU, S. CHEN, G. BORMAN, P. MYERS, and O. UYEHARA (1965), ‘‘Development and
Evaluation of the Simulation of the CI Engine,’’ SAE paper 650451.
MOTOYOSHI, E., T. YAMADA, and M. MORI (1976), ‘‘The Combustion and Exhaust Emission Characteristics and Starting Ability of Y.P.C. Combustion System,’’ SAE paper 760215.
PISCHINGER, R. and W. CARTELLIERI (1972), ‘‘Combustion System Parameters and Their Effect upon
Diesel Engine Exhaust Emissions, ‘‘SAE paper 720756.
ROENSCH, M. (1949), ‘‘Thermal Efficiency and Mechanical Losses of Automotive Engines,’’ SAE J.,
Vol. 51, p. 1730.
Homework
385
SHAYLER, P., J. CHICK, and D. EADE (1999), ‘‘A Method of Predicting Brake Specific Fuel Consumption
Maps,’’ SAE paper 1999-01-0556.
THOMAS, F. J., J. S. AHLUWALIA, E. SHAMAH, and G. W. VAN DER HORST (1984), ‘‘Medium-Speed
Diesel Engines Part 1: Design Trends and the Use of Residual/Blended Fuels,’’ ASME paper
84-DGP-15.
WALDER, C. J. (1965), ‘‘Problems in the Design and Development of High Speed Diesel Engines,’’
SAE paper 978A.
YOUNG, M. B. (1980), ‘‘Cyclic Dispersion-Some Quantitative Cause and Effect Relationships,’’ SAE
paper 800459.
13.10 HOMEWORK
13.1 Derive Equation 13.1, which relates the power of an engine to its volumetric efficiency,
net indicated thermal efficiency, mechanical efficiency, and engine speed.
13.2 Derive Equation 13.1 as a function of mean piston speed 𝑈̄ p , instead of engine speed.
13.3 What is the specific brake work (kJ/kgfuel ) of a 4 L single-cylinder propane engine operating
stoichiometrically, if the heat transfer to the coolant is 17 kW, the air and fuel enter the
engine at 298 K, the exhaust is 700 K, and the propane mass flow rate is 1.2 g/s?
13.4 A hydrogen engine operates with an air mass flowrate of 2.0 kg/s and produces 1 MW of
power. The exhaust enthalpy is equal to −40,000 kJ/kgfuel , and the heat losses from the
engine total 50,000 kJ/kmolfuel . At what equivalence ratio 𝜙 is the engine being operated?
13.5 A four-stroke 6 L engine is fueled lean with methane at an equivalence ratio 𝜙 = 0.8. It
operates at 2000 rpm with a volumetric efficiency of 0.80. The exhaust temperature is
800 K, and the heat transfer to the coolant is 3.4 ×105 kJ/kmolfuel . What is the engine’s
thermal efficiency and power?
13.6 A car traveling steadily on a level road at 100 km/h requires about 15 kW of power from
the engine. For the engine families represented by the performance maps of Figure 13.6
and Figure 13.7, estimate the fuel economy of the vehicle (km/g), the bore required of a
four-cylinder engine, and the maximum power the engine will produce. Assume that the
engine is operating at its best fuel economy point when the vehicle is cruising at 100 km/h,
that engine controls limit the piston speed to 10 m/s, and that the performance map is size
independent.
13.7 Using the performance map of Figure 13.6, calculate the required cylinder bore, for a
six-cylinder engine with equal bore and stroke that is to produce 200 kW with a maximum
piston speed of 12 m/s. Plot the bsfc, torque, and power versus engine speed.
13.8 A six-cylinder diesel engine with equal bore and stroke is being designed to provide a
maximum brake torque of 200 Nm at 2000 rpm. Using the performance map of Figure
13.7, estimate the required engine displacement, and maximum brake power.
13.9 What is the engine power required for an automobile to travel up a hill with a 10◦ slope at
50 mph? Assume a vehicle frontal cross-sectional area 𝐴v of 2.0 m2 , 𝐶d = 0.3, 𝐶r = 0.015,
𝑚v = 1500 kg.
13.10 If the power required for a truck to travel up a 12◦ incline at 70 mph is 91 kW, what is the
mass 𝑚 of the truck? The frontal area is 2.5 m2 , 𝐶d = 0.5, and 𝐶r = 0.02.
386
Overall Engine Performance
13.11 The price of large diesel engines is roughly proportional to their rated power. Let 𝑐1 be the
engine price per kilowatt per year and 𝑐2 the fuel price per kilogram. At low values of 𝑐1 ,
it pays to buy an engine bigger than required and operate it at its best fuel economy point.
For low values of 𝑐2 , it pays to buy a smaller engine and run it at its rated power. For the
diesel engine family performance map of Figure 13.8, at what ratio 𝑐1 ∕𝑐2 will two different
sized engines yield the same total annual cost? Assume the engines are run 20 h/day and
their rated power is at a speed-load point of 𝑈p = 8 m/s and bmep = 8 bar.
13.12 Write an expression resembling Equation 13.14 for the mass of pollutant species 𝑖 (given
its emission index at any load--speed point) emitted by an engine operated over a duty cycle
from 0 < 𝑡 < 𝑡d .
Appendix
A
Physical Properties of Air
Figure A.1 Specific heat ratio
for air
Table A.1 Properties of Air at Atmospheric Pressure
T
(K)
𝜌
(kg/m3 )
𝑐𝑝
(kJ/(kg K))
𝜇 × 107
(N s/m2 )
𝜈 × 106
m2 /s2
𝑘 × 103
(W/(m K))
𝛼 × 106
(m2 /s2 )
𝑃𝑟
100
150
200
250
300
350
400
450
500
550
3.5562
2.3364
1.7458
1.3947
1.1614
0.9950
0.8711
0.7740
0.6964
0.6329
1.032
1.012
1.007
1.006
1.007
1.009
1.014
1.021
1.030
1.040
71.1
103.4
132.5
159.6
184.6
208.2
230.1
250.7
270.1
288.4
2.00
4.426
7.590
11.44
15.89
20.92
26.41
32.39
38.79
45.57
9.34
13.8
18.1
22.3
26.3
30.0
33.8
37.3
40.7
43.9
2.54
5.84
10.3
15.9
22.5
29.9
38.3
47.2
56.7
66.7
0.786
0.758
0.737
0.720
0.707
0.700
0.690
0.686
0.684
0.683
(continued)
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
387
388
Physical Properties of Air
Table A.1
(Continued)
T
(K)
𝜌
(kg/m3 )
𝑐𝑝
(kJ/(kg K))
𝜇 × 107
(N s/m2 )
𝜈 × 106
m2 /s2
𝑘 × 103
(W/(m K))
600
650
700
750
800
850
900
950
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
3000
0.5804
0.5356
0.4975
0.4643
0.4354
0.4097
0.3868
0.3666
0.3482
0.3166
0.2902
0.2679
0.2488
0.2322
0.2177
0.2049
0.1935
0.1833
0.1741
0.1658
0.1582
0.1513
0.1448
0.1389
0.1135
1.051
1.063
1.075
1.087
1.099
1.110
1.121
1.131
1.141
1.159
1.175
1.189
1.207
1.230
1.248
1.267
1.286
1.307
1.337
1.372
1.417
1.478
1.558
1.665
2.726
305.8
322.5
338.8
354.6
369.8
384.3
398.1
411.3
424.4
449.0
473.0
496.0
530
557
584
611
637
663
689
715
740
766
792
818
955
52.69
60.21
68.10
76.37
84.93
93.80
102.9
112.2
121.9
141.8
162.9
185.1
213
240
268
298
329
362
396
431
468
506
547
589
841
46.9
49.7
52.4
54.9
57.3
59.6
62.0
64.3
66.7
71.5
76.3
82
91
100
106
113
120
128
137
147
160
175
196
222
486
𝛼 × 106
(m2 /s2 )
76.9
87.3
98.0
109
120
131
143
155
168
195
224
238
303
350
390
435
482
534
589
646
714
783
869
960
1570
𝑃𝑟
0.685
0.690
0.695
0.702
0.709
0.716
0.720
0.723
0.726
0.728
0.728
0.719
0.703
0.685
0.688
0.685
0.683
0.677
0.672
0.667
0.655
0.647
0.630
0.613
0.536
Source: F. Incropera, D. DeWitt, T. Bergman, and A. Lavine (2007), Fundamentals of Heat and Mass Transfer,
Wiley, New York.
Table A.2 Physical Properties of Air at Atmospheric Conditions
(𝑇 = 298 K, 𝑃 = 1 atm = 1.0133 bar)
Molecular mass
Gas constant
Speed of sound
Binary diffusion with octane
Schmidt number
Lewis number
𝑀 = 28.966 kg/kmol
𝑅 = 0.28704 kJ/(kg K)
𝑐 = 345.9 m/s
𝐷 = 5.68 × 10−6 m2 /s
𝑆𝑐 = 𝜈∕𝐷 = 2.77
𝐿𝑒 = 𝛼∕𝐷 = 3.91
Appendix
B
Thermodynamic Property
Tables for Various Ideal
Gases
Table B.1 Properties of Various Ideal Gases at 298 K (SI Units)
Gas
Chemical
formula
𝑀
(kg/kmol)
𝑅
(kJ/(kg K))
𝑐po
(kJ/(kg K))
𝑐vo
(kJ/(kg K))
𝛾
Air
Ethane
Ethanol
Ethylene
Helium
Hydrogen
Methane
Methanol
Nitrogen
Nitrous oxide
𝑛-Octane
Oxygen
Propane
Steam
C2 H6
C2 H5 OH
C2 H4
He
H2
CH4
CH3 OH
N2
N2 O
C8 H18
O2
C3 H8
H2 O
28.97
30.07
46.069
28.054
4.003
2.016
16.04
32.042
28.013
44.013
114.23
31.999
44.097
18.015
0.287
0.27650
0.18048
0.29637
2.07703
4.12418
0.51835
0.25948
0.29680
0.18891
0.07279
0.25983
0.18855
0.46152
1.004
1.7662
1.427
1.5482
5.1926
14.2091
2.2537
1.4050
1.0416
0.8793
1.7113
0.9216
1.6794
1.8723
0.717
1.4897
1.246
1.2518
3.1156
10.0849
1.7354
1.1455
0.7448
0.6904
1.6385
0.6618
1.4909
1.4108
1.40
1.186
1.145
1.237
1.667
1.409
1.299
1.227
1.400
1.274
1.044
1.393
1.126
1.327
Source: R. Sonntag, C. Borgnakke, and G. Van Wylen (2003), Fundamentals of Thermodynamics, Wiley,
New York.
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
389
390
Thermodynamic Property Tables for Various Ideal Gases
Table B.2 Binary Diffusion Coefficients at 1 atm (𝐷AB ∼ 𝑇 3∕2 ∕𝑃 )
Substance A
Substance B
Benzene
Carbon dioxide
Cyclohexane
𝑛-Decane
𝑛-Dodecane
Ethanol
𝑛-Hexane
Hydrogen
Methanol
𝑛-Octane
Water
Air
Air
Air
Nitrogen
Nitrogen
Air
Nitrogen
Air
Air
Air
Air
𝑇 (K)
𝐷AB × 105 (m2 /s)
273
273
318
363
399
273
288
273
273
273
273
0.77
1.38
0.86
0.84
0.81
1.02
0.757
0.611
1.32
0.505
2.2
Source: Perry, Green, and Maloney (1984), Perry’s Chemical Engineers’ Handbook, McGraw-Hill, New York.
Table B.3 Ideal Gas Properties of N2 and N (SI Units), Entropies at 0.1-MPa (1-bar) Pressure
Nitrogen, Diatomic (N2 )
ℎ̄ of,298 = 0 kJ/kmol
𝑀 = 28.013
𝑇 (K)
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
0
100
200
298
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
−8670
−5768
−2857
0
54
2971
5911
8894
11,937
15,046
18,223
21,463
24,760
28,109
31,503
34,936
38,405
41,904
45,430
48,979
52,549
56,137
𝑠̄o
(kJ/(kmol K))
0
159.812
179.985
191.609
191.789
200.181
206.740
212.177
216.865
221.016
224.757
228.171
231.314
234.227
236.943
239.487
241.881
244.139
246.276
248.304
250.234
252.075
Nitrogen, Monatomic (N)
ℎ̄ of,298 = 472,680 kJ/kmol
𝑀 = 14.007
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
𝑠̄o
(kJ/(kmol K))
−6197
−4119
−2040
0
38
2117
4196
6274
8353
10,431
12,510
14,589
16,667
18,746
20,825
22,903
24,982
27,060
29,139
31,218
33,296
35,375
0
130.593
145.001
153.300
153.429
159.409
164.047
167.837
171.041
173.816
176.265
178.455
180.436
182.244
183.908
185.448
186.883
188.224
189.484
190.672
191.796
192.863
(continued)
Thermodynamic Property Tables for Various Ideal Gases
Table B.3
391
(Continued)
Nitrogen, Diatomic (N2 )
ℎ̄ of,298 = 0 kJ/kmol
𝑀 = 28.013
𝑇 (K)
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
2200
2400
2600
2800
3000
3200
3400
3600
3800
4000
4400
4800
5200
5600
6000
63,362
70,640
77,963
85,323
92,715
100,134
107,577
115,042
122,526
130,027
145,078
160,188
175,352
190,572
205,848
𝑠̄o
(kJ/(kmol K))
255.518
258.684
261.615
264.342
266.892
269.286
271.542
273.675
275.698
277.622
281.209
284.495
287.530
290.349
292.984
Nitrogen, Monatomic (N)
ℎ̄ of,298 = 472,680 kJ/kmol
𝑀 = 14.007
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
39,534
43,695
47,860
52,033
56,218
60,420
64,646
68,902
73,194
77,532
86,367
95,457
104,843
114,550
124,590
𝑠̄o
(kJ/(kmol K))
194.845
196.655
198.322
199.868
201.311
202.667
203.948
205.164
206.325
207.437
209.542
211.519
213.397
215.195
216.926
Source: R. Sonntag, C. Borgnakke, and G. Van Wylen (2003), Fundamentals of Thermodynamics, John Wiley,
New York.
392
Thermodynamic Property Tables for Various Ideal Gases
Table B.4 Ideal Gas Properties of O2 and O (SI Units), Entropies at 0.1-MPa (1-bar) Pressure
Oxygen, Diatomic (O2 )
ℎ̄ o = 0 kJ/kmol
f,298
𝑀 = 31.999
𝑇 (K)
0
100
200
298
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2200
2400
2600
2800
3000
3200
3400
3600
3800
4000
4400
4800
5200
5600
6000
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
−8683
−5777
−2868
0
54
3027
6086
9245
12,499
15,836
19,241
22,703
26,212
29,761
33,345
36,958
40,600
44,267
47,959
51,674
55,414
59,176
66,770
74,453
82,225
90,080
98,013
106,022
114,101
122,245
130,447
138,705
155,374
172,240
189,312
206,618
224,210
𝑠̄o
(kJ/(kmol K))
0
173.308
193.483
205.148
205.329
213.873
220.693
226.450
231.465
235.920
239.931
243.579
246.923
250.011
252.878
255.556
258.068
260.434
262.673
264.797
266.819
268.748
272.366
275.708
278.818
281.729
284.466
287.050
289.499
291.826
294.043
296.161
300.133
303.801
307.217
310.423
313.457
Oxygen, Monatomic (O)
ℎ̄ of,298 = 249,170 kJ/kmol
𝑀 = 16.00
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
−6725
−4518
−2186
0
41
2207
4343
6462
8570
10,671
12,767
14,860
16,950
19,039
21,126
23,212
25,296
27,381
29,464
31,547
33,630
35,713
39,878
44,045
48,216
52,391
56,574
60,767
64,971
69,190
73,424
77,675
86,234
94,873
103,592
112,391
121,264
𝑠̄o
(kJ/(kmol K))
0
135.947
152.153
161.059
161.194
167.431
172.198
176.060
179.310
182.116
184.585
186.790
188.783
190.600
192.270
193.816
195.254
196.599
197.862
199.053
200.179
201.247
203.232
205.045
206.714
208.262
209.705
211.058
212.332
213.538
214.682
215.773
217.812
219.691
221.435
223.066
224.597
Source: R. Sonntag, C. Borgnakke, and G. Van Wylen (2003), Fundamentals of Thermodynamics, John Wiley,
New York.
Thermodynamic Property Tables for Various Ideal Gases
393
Table B.5 Ideal Gas Properties of CO2 and CO (SI Units), Entropies at 0.1-MPa (1-bar) Pressure
Carbon Dioxide (CO2 )
ℎ̄ of,298 = −393,522 kJ/kmol
𝑀 = 44.01
𝑇 (K)
0
100
200
298
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2200
2400
2600
2800
3000
3200
3400
3600
3800
4000
4400
4800
5200
5600
6000
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
−9364
−6457
−3413
0
69
4003
8305
12,906
17,754
22,806
28,030
33,397
38,885
44,473
50,148
55,895
61,705
67,569
73,480
79,432
85,420
91,439
103,562
115,779
128,074
140,435
152,853
165,321
177,836
190,394
202,990
215,624
240,992
266,488
292,112
317,870
343,782
𝑠̄o
(kJ/(kmol K))
0
179.010
199.976
213.794
214.024
225.314
234.902
243.284
250.752
257.496
263.646
269.299
274.528
279.390
283.931
288.190
292.199
295.984
299.567
302.969
306.207
309.294
315.070
320.384
325.307
329.887
334.170
338.194
341.988
345.576
348.981
352.221
358.266
363.812
368.939
373.711
378.180
Carbon Monoxide (CO)
ℎ̄ of,298 = −110,527 kJ/kmol
𝑀 = 28.01
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
−8671
−5772
−2860
0
54
2977
5932
8942
12,021
15,174
18,397
21,686
25,031
28,427
31,867
35,343
38,852
42,388
45,948
49,529
53,128
56,743
64,012
71,326
78,679
86,070
93,504
100,962
108,440
115,938
123,454
130,989
146,108
161,285
176,510
191,782
207,105
𝑠̄o
(kJ/(kmol K))
0
165.852
186.024
197.651
197.831
206.240
212.833
218.321
223.067
227.277
231.074
234.538
237.726
240.679
243.431
246.006
248.426
250.707
252.866
254.913
256.860
258.716
262.182
265.361
268.302
271.044
273.607
276.012
278.279
280.422
282.454
284.387
287.989
291.290
294.337
297.167
299.809
Source: R. Sonntag, C. Borgnakke, and G. Van Wylen (2003), Fundamentals of Thermodynamics, John Wiley,
New York.
394
Thermodynamic Property Tables for Various Ideal Gases
Table B.6 Ideal Gas Properties of H2 O and OH (SI Units), Entropies at 0.1-MPa (1-bar) Pressure
Water (H2 O)
ℎ̄ of,298 = −241,826 kJ/kmol
𝑀 = 18.015
𝑇 (K)
0
100
200
298
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2200
2400
2600
2800
3000
3200
3400
3600
3800
4000
4400
4800
5200
5600
6000
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
−9904
−6617
−3282
0
62
3450
6922
10,499
14,190
18,002
21,937
26,000
30,190
34,506
38,941
43,491
48,149
52,907
57,757
62,693
67,706
72,788
83,153
93,741
104,520
115,463
126,548
137,756
149,073
160,484
171,981
183,552
206,892
230,456
254,216
278,161
302,295
𝑠̄o
(kJ/(kmol K))
0
152.386
175.488
188.835
189.043
198.787
206.532
213.051
218.739
223.826
228.460
232.739
236.732
240.485
244.035
247.406
250.620
253.690
256.631
259.452
262.162
264.769
269.706
274.312
278.625
282.680
286.504
290.120
293.550
296.812
299.919
302.887
308.448
313.573
318.328
322.764
326.926
Hydroxyl (OH)
ℎ̄ of,298 = 38,987 kJ/kmol
𝑀 = 17.007
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
−9172
−6140
−2975
0
55
3034
5991
8943
11,902
14,881
17,889
20,935
24,024
27,159
30,340
33,567
36,838
40,151
43,502
46,890
50,311
53,763
60,751
67,840
75,018
82,268
89,585
96,960
104,388
111,864
119,382
126,940
142,165
157,522
173,002
188,598
204,309
𝑠̄o
(kJ/(kmol K))
0
149.591
171.592
183.709
183.894
192.466
199.066
204.448
209.008
212.984
216.526
219.735
222.680
225.408
227.955
230.347
232.604
234.741
236.772
238.707
240.556
242.328
245.659
248.743
251.614
254.301
256.825
259.205
261.456
263.592
265.625
267.563
271.191
274.531
277.629
280.518
283.227
Source: R. Sonntag, C. Borgnakke, and G. Van Wylen (2003), Fundamentals of Thermodynamics, John Wiley,
New York.
Thermodynamic Property Tables for Various Ideal Gases
395
Table B.7 Ideal Gas Properties of H2 and H (SI Units), Entropies at 0.1-MPa (1-bar) Pressure
Hydrogen (H2 )
ℎ̄ of,298 = 0 kJ/kmol
𝑀 = 2.016
𝑇 (K)
0
100
200
298
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2200
2400
2600
2800
3000
3200
3400
3600
3800
4000
4400
4800
5200
5600
6000
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
−8467
−5467
−2774
0
53
2961
5883
8799
11,730
14,681
17,657
20,663
23,704
26,785
29,907
33,073
36,281
39,533
42,826
46,160
49,532
52,942
59,865
66,915
74,082
81,355
88,725
96,187
103,736
111,367
119,077
126,864
142,658
158,730
175,057
191,607
208,332
𝑠̄o
(kJ/(kmol K))
0
100.727
119.410
130.678
130.856
139.219
145.738
151.078
155.609
159.554
163.060
166.225
169.121
171.798
174.294
176.637
178.849
180.946
182.941
184.846
186.670
188.419
191.719
194.789
197.659
200.355
202.898
205.306
207.593
209.773
211.856
213.851
217.612
221.109
224.379
227.447
230.322
Hydrogen, Monatomic (H)
ℎ̄ of,298 = 217,999 kJ/kmol
𝑀 = 1.008
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
−6197
−4119
−2040
0
38
2117
4196
6274
8353
10,431
12,510
14,589
16,667
18,746
20,825
22,903
24,982
24,060
29,139
31,218
33,296
35,375
39,532
43,689
47,847
52,004
56,161
60,318
64,475
68,633
72,790
76,947
85,261
93,576
101,890
110,205
118,519
𝑠̄o
(kJ/(kmol K))
0
92.009
106.417
114.716
114.845
120.825
125.463
129.253
132.457
135.233
137.681
139.871
141.852
143.661
145.324
146.865
148.299
149.640
150.900
152.089
153.212
154.279
156.260
158.069
159.732
161.273
162.707
164.048
165.308
166.497
167.620
168.687
170.668
172.476
174.140
175.681
177.114
Source: R. Sonntag, C. Borgnakke, and G. Van Wylen (2003), Fundamentals of Thermodynamics, John Wiley,
New York.
396
Thermodynamic Property Tables for Various Ideal Gases
Table B.8 Ideal Gas Properties of NO and NO2 (SI Units), Entropies at 0.1-MPa (1-bar) Pressure
Nitric Oxide (NO)
ℎ̄ of,298 = 90,291 kJ/kmol
𝑀 = 30.006
𝑇 (K)
0
100
200
298
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2200
2400
2600
2800
3000
3200
3400
3600
3800
4000
4400
4800
5200
5600
6000
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
−9192
−6073
−2951
0
55
3040
6059
9144
12,308
15,548
18,858
22,229
25,653
29,120
32,626
36,164
39,729
43,319
46,929
50,557
54,201
57,859
65,212
72,606
80,034
87,491
94,973
102,477
110,000
117,541
125,099
132,671
147,857
163,094
178,377
193,703
209,070
𝑠̄o
(kJ/(kmol K))
0
177.031
198.747
210.759
210.943
219.529
226.263
231.886
236.762
241.088
244.985
248.536
251.799
254.816
257.621
260.243
262.703
265.019
267.208
269.282
271.252
273.128
276.632
279.849
282.822
285.585
288.165
290.587
292.867
295.022
297.065
299.007
302.626
305.940
308.998
311.838
314.488
Nitrogen Dioxide (NO2 )
ℎ̄ of,298 = 33,100 kJ/kmol
𝑀 = 46.005
(ℎ̄ − ℎ̄ o298 )
(kJ/kmol)
−10186
−6861
−3495
0
68
3927
8099
12,555
17,250
22,138
27,180
32,344
37,606
42,946
48,351
53,808
59,309
64,846
70,414
76,008
81,624
87,259
98,578
109,948
121,358
132,800
144,267
155,756
167,262
178,783
190,316
201,860
224,973
248,114
271,276
294,455
317,648
𝑠̄o
(kJ/(kmol K))
0
202.563
225.852
240.034
240.263
251.342
260.638
268.755
275.988
282.513
288.450
293.889
298.904
303.551
307.876
311.920
315.715
319.289
322.664
325.861
328.898
331.788
337.182
342.128
346.695
350.934
354.890
358.597
362.085
365.378
368.495
371.456
376.963
381.997
386.632
390.926
394.926
Source: R. Sonntag, C. Borgnakke, and G. Van Wylen (2003), Fundamentals of Thermodynamics, John Wiley,
New York.
Appendix
C
Curve-Fit Coefficients for
Thermodynamic Properties
of Various Fuels and Ideal Gases
Specific heats of fuels and ideal gases are curve-fitted to polynomials of the form below.
For any given species, the specific heat is approximated by
𝑐𝑝
𝑅
=
𝑐̄𝑝
𝑅u
= 𝑎 1 + 𝑎 2 𝑇 + 𝑎 3 𝑇 2 + 𝑎4 𝑇 3 + 𝑎 5 𝑇 4
For an ideal gas, 𝑑ℎ = 𝑐𝑝 𝑑𝑇 and 𝑑𝑠 = (𝑐𝑝 ∕𝑇 )𝑑𝑇 . It follows that the enthalpy and
entropy at atmospheric pressure are
𝑎
𝑎
𝑎
𝑎
𝑎
ℎ̄
ℎ
=
= 𝑎1 + 2 𝑇 + 3 𝑇 2 + 4 𝑇 3 + 5 𝑇 4 + 6
𝑅𝑇
𝑅u 𝑇
2
3
4
5
𝑇
𝑎
𝑎
𝑎
𝑠̄o
𝑠o
= 𝑎1 ln 𝑇 + 𝑎2 𝑇 + 3 𝑇 2 + 4 𝑇 3 + 5 𝑇 4 + 𝑎7
=
𝑅
𝑅u
2
3
4
The temperature range 300--1000 K is useful for unburned mixture property computation, and the temperature range 1000--3000 K is useful for burned mixture property
computation.
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
397
Appendix
D
Conversion Factors and
Physical Constants
Table D.1 Unit Conversion Factors
Area
1 m2
Energy
Energy/mass
Force
Heat transfer rate
Heat flux
Heat transfer coefficient
Thermal diffusivity
Length
1J
1kJ/kg
1N
1W
1 W/m2
1 W/(m2 -K)
1 m2 /s
1m
Mass
Mass density
Mass flow rate
Mass transfer coefficient
Power
Pressure and stress
1 km
1 kg
1 kg/m3
1 kg/s
1 m/s
1 kW
1 Pa (1 N/m2 )
1.0133 ×105 N/m2
1 ×105 N/m2
Rotational speed
Specific heat
Temperature
1 rev/min
1 J/(kg-K)
K
Temperature difference
1K
= 1550.0 in.2
= 10.764 ft2
= 9.4787 ×10−4 Btu
= 0.4303 Btu/lbm
= 0.22481 lbf
= 3.4123 Btu/h
= 0.3171 Btu/(h-ft2 )
= 0.17612 Btu/(h-ft2 ◦ F)
= 3.875 ×104 ft2 /h
= 39.370 in.
= 3.2808 ft
= 0.62137 mile
= 2.2046 lbm
= 0.062428 lbm /ft3
= 7936.6 lbm /h
= 1.1811×104 ft/h
= 1.341 hp
= 0.020886 lbf /ft2
= 1 standard atmosphere
= 760 mmHg
= 1 bar
= 750.06 mmHg
= 0.10472 rad/s
= 2.3886 ×10−4 Btu/(lbm -◦ F)
= (5/9)◦ R
= (5/9) (◦ R +459.67)
◦
C +273.15
= 1◦ C
= (9/5)◦ R = (9/5)◦ F
(continued)
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
401
402
Conversion Factors and Physical Constants
Table D.1
(Continued)
Thermal conductivity
Thermal resistance
Torque
Viscosity (dynamic)
Viscosity (kinematic)
Volume
1 W/(m-K)
1 K/W
1 Nm
1 kg/(m-s2 )
1 m2 /s
1 m3 (103 L)
Volume flow rate
1 m3 /s
= 0.57782 Btu/(h ft-◦ F)
= 0.52750◦ F/(h-Btu)
= 0.73756 lbf ft
= 2419.1 lbm /(ft-h)
= 3.875 ×104 ft2 /h
= 6.1023×104 in.3
= 35.314 ft3
= 264.17 gal
= 2.1189 ×103 ft3 /min
= 1.5850 ×104 gal/min
Source: F. Incropera, D. DeWitt, T. Bergman, and A. Lavine, (2007), Fundamentals of Heat and Mass Transfer,
John Wiley, New York.
Table D.2 Physical Constants
Universal gas constant
𝑅u
Avogadro’s number
Planck’s constant
Boltzmann’s constant
Speed of light in vacuum
Stefan--Boltzmann constant
𝑁
ℎ
𝑘
𝑐o
𝜎
Gravitational acceleration
Standard atmospheric pressure
𝑔
𝑃
= 8.315 kJ/(kmol-K)
= 8.314 × 10−2 m3 bar/(kmol-K)
= 8.205 × 10−2 m3 atm/(kmol-K)
= 1545 ft lbf /(lbmole-◦ R)
= 1.986 Btu/(lbmole-◦ R)
= 6.024 ×1023 molecules/mol
= 6.625 ×10−34 (J-s)/molecule
= 1.380 ×10−23 J/(K-molecule)
= 2.998 ×108 m/s
= 5.670 ×10−8 W/(m2 -K4 )
= 0.1714 ×10−8 Btu/(h ft2 -◦ R4 )
= 9.807 m/s2
= 101,325 N/m2
= 101.325 kPa
Appendix
E
Thermodynamic Analysis
of Mixtures
E.1 THERMODYNAMIC DERIVATIVES
As shown by Bridgman (1914), any thermodynamic derivative can be expressed in terms
of three independent derivatives. These derivatives are
( )
𝜕ℎ
= 𝑐𝑝
( 𝜕𝑇 )
𝜕𝑣
(E.1)
𝜕𝑃
( )𝑇
𝜕𝑣
𝜕𝑇 𝑃
For example, 𝛾, the ratio of specific heats, is given by
𝑐𝑝
𝑐𝑝
𝛾=
=
(E.2)
𝑐𝑣
𝑐𝑝 + 𝑇 (𝜕𝑣∕𝜕𝑇 )2𝑃 ∕ (𝜕𝑣∕𝜕𝑃 )𝑇
These derivatives were chosen by Bridgman as they can be determined directly by
experiment. The term (𝜕𝑣∕𝜕𝑃 )𝑇 is the isothermal compressibility, the term (𝜕𝑣∕𝜕𝑇 )𝑃 is
the coefficient of volumetric expansion, and the term (𝜕ℎ∕𝜕𝑇 ) is the specific heat at constant
pressure. These three variables are used in this text to compute fuel--air mixture properties
and perform cycle analysis.
For a gas mixture, the derivatives can be expressed as functions of pressure 𝑃 , specific
volume 𝑣, temperature 𝑇 , and molecular mass 𝑀. Using the ideal gas law, the specific
volume 𝑣 of a mixture is
𝑅 𝑇
(E.3)
𝑣= u
𝑀𝑃
Upon differentiation of Equation E.3 with respect to pressure 𝑃 ,
( )
𝑅 𝑇 𝜕 ( 1 ) 𝑅u 𝑇 𝜕 ( 1 )
𝜕𝑣
+
= u
𝜕𝑃 𝑇
𝑀 𝜕𝑃 𝑃
𝑃 𝜕𝑃 𝑀
𝑣
𝑣 𝜕𝑀
=− −
(E.4)
𝑃
𝑀 𝜕𝑃
𝜕𝑦
𝑣 ∑
𝑣
𝑀𝑖 𝑖
=− −
𝑃
𝑀
𝜕𝑃
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
403
404
Thermodynamic Analysis of Mixtures
where
𝜕𝑦
𝜕𝑀 ∑
=
𝑀𝑖 𝑖
𝜕𝑃
𝜕𝑃
and upon differentiation of Equation E.3 with respect to temperature 𝑇 ,
( )
𝑅
𝑅 𝑇 𝜕 ( 1 )
𝜕𝑣
= u + u
𝜕𝑇 𝑃
𝑃𝑀
𝑃 𝜕𝑇 𝑀
𝑣
𝑣 𝜕𝑀
= −
𝑇
𝑀 𝜕𝑇
𝜕𝑦
𝑣 ∑
𝑣
𝑀𝑖 𝑖
=− −
𝑇
𝑀
𝜕𝑇
(E.5)
(E.6)
where
𝜕𝑦
𝜕𝑀 ∑
=
𝑀𝑖 𝑖
𝜕𝑇
𝜕𝑇
(E.7)
Note that the important role of the changes in mole fraction with respect to pressure and
temperature, 𝜕𝑦𝑖 ∕𝜕𝑃 and 𝜕𝑦𝑖 ∕𝜕𝑇 , have in determination of the mixture properties and
thermodynamic derivatives.
Adopting the notation used by Bridgeman (1914),
(
𝜕𝑦
𝜕𝑥
)
=
𝑧
(𝜕𝑦)𝑧
(𝜕𝑥)𝑧
the following list can be used to construct any other required thermodynamic first derivative
in terms of 𝑐𝑝 = 𝜕ℎ∕𝜕𝑇 , 𝜕𝑣∕𝜕𝑃 , and 𝜕𝑣∕𝜕𝑇 :
(𝜕𝑇 )𝑃 = − (𝜕𝑃 )𝑇 = 1
( )
𝜕𝑣
(𝜕𝑣)𝑃 = − (𝜕𝑃 )𝑣 =
𝜕𝑇 𝑃
𝑐𝑝
(𝜕𝑠)𝑃 = − (𝜕𝑃 )𝑠 =
𝑇
( )
𝜕𝑣
(𝜕𝑢)𝑃 = − (𝜕𝑃 )u = 𝑐𝑝 − 𝑃
𝜕𝑇 𝑃
(𝜕ℎ)𝑃 = − (𝜕𝑃 )ℎ = 𝑐𝑝
( )
𝜕𝑣
(𝜕𝑣)𝑇 = − (𝜕𝑇 )𝑣 = −
𝜕𝑃 𝑇
( )
𝜕𝑣
(𝜕𝑠)𝑇 = − (𝜕𝑇 )𝑠 =
𝜕𝑇 𝑃
( )
( )
𝜕𝑣
𝜕𝑣
+𝑃
(𝜕𝑢)𝑇 = − (𝜕𝑇 )u = 𝑇
𝜕𝑇 𝑃
𝜕𝑃 𝑇
( )
𝜕𝑣
(𝜕ℎ)𝑇 = − (𝜕𝑇 )ℎ = −𝑣 + 𝑇
𝜕𝑇 𝑃
( )2
𝑐𝑝 ( 𝜕𝑣 )
𝜕𝑣
+
(𝜕𝑠)𝑣 = − (𝜕𝑣)𝑠 =
𝑇 𝜕𝑃 𝑇
𝜕𝑇 𝑃
( )
( )2
𝜕𝑣
𝜕𝑣
+𝑇
(𝜕𝑢)𝑣 = − (𝜕𝑣)u = 𝑐𝑝
𝜕𝑃 𝑇
𝜕𝑇 𝑃
Thermodynamic Analysis of Mixtures
405
)
( )2
( )
𝜕𝑣
𝜕𝑣
𝜕𝑣
+𝑇
−𝑣
𝜕𝑃 𝑇
𝜕𝑇 𝑃
𝜕𝑇 𝑃
[ ( )
( )2 ]
𝜕𝑣
𝜕𝑣
𝑃
+𝑇
𝑐
(𝜕𝑢)𝑠 = − (𝜕𝑠)u =
𝑇 𝑝 𝜕𝑃 𝑇
𝜕𝑇 𝑃
𝑣𝑐𝑝
(𝜕ℎ)𝑠 = − (𝜕𝑠)ℎ = −
𝑇
[ ( )
( )2 ]
[
( ) ]
𝜕𝑣
𝜕𝑣
𝜕𝑣
− 𝑃 𝑐𝑝
+𝑇
(𝜕ℎ)u = − (𝜕𝑢)ℎ = −𝑣 𝑐𝑝 − 𝑃
𝜕𝑇 𝑃
𝜕𝑃 𝑇
𝜕𝑇 𝑃
(𝜕ℎ)𝑣 = − (𝜕𝑣)ℎ = 𝑐𝑝
(
E.2 NUMERICAL SOLUTION OF EQUILIBRIUM COMBUSTION
EQUATIONS
The 4 × 4 nonlinear system of equations developed in Chapter 3.7 with unknowns
𝑦3 , 𝑦4 , 𝑦5 , 𝑦6 , that is, the mole fractions of N2 , O2 , CO, H2 , respectively, is solved using Newton--Raphson iteration and Gaussian elimination.
𝑓𝑗 (𝑦3 , 𝑦4 , 𝑦5 , 𝑦6 ) = 0 𝑗 = 1, 2, 3, 4
(E.8)
A first-order Taylor series expansion of the solution mole fraction vector [𝑦∗3 , 𝑦∗4 , 𝑦∗5 , 𝑦∗6 ] is
∗
𝑦∗𝑖 = 𝑦(1)
𝑖 + Δ𝑦𝑖 , where 𝑦𝑖 is a set of mole fractions reasonably close to the solution vector.
The steps Δ𝑦𝑖 are computed from the matrix equation
(E.9)
𝐀Δ𝑦𝑖 = 𝐅
where 𝐀 is the Jacobian matrix of the partial derivative of each equation with respect to
each mole fraction, 𝐴𝑖𝑗 = 𝜕𝑓𝑗 ∕𝜕𝑦𝑖 . At each iteration denoted by 𝑘, the new 𝑦𝑖 solution
vector is thus given by
= 𝑦𝑘𝑖 − 𝐀−1 𝐅
𝑦𝑘+1
𝑖
(E.10)
Iteration continues until the maximum step change Δ𝑦𝑖 is below a specified tolerance. In
certain cases, Newton--Raphson iteration may run into difficulties due to the numerical
precision of the computer, and the solution of the matrix equation for Δ𝑦𝑖 may yield
singularities. This behavior is reduced by introducing an underrelaxation parameter, which
was determined by trial and error to be 0.05.
The elements of the Jacobian matrix 𝐀 are determined by first defining the following
partial derivatives:
𝐷𝑖𝑗 =
𝐷𝑗𝑗 =
𝜕𝑦𝑘 𝑖 = 1, 2, 7, 8, 9, 10
𝜕𝑦𝑙 𝑗 = 3, 4, 5, 6
𝜕𝑦𝑗
𝜕𝑦𝑗
=1
Performing the mole fraction differentiations yields
𝐷76 =
1 𝑐1
2 𝑦1∕2
1∕2
𝐷103 =
6
𝐷84
1 𝑐2
=
2 𝑦1∕2
4
1 𝑐4 𝑦4
2 𝑦1∕2
3
𝐷26 =
1∕2
𝑐5 𝑦4
406
Thermodynamic Analysis of Mixtures
1∕2
𝐷94
1 𝑐3 𝑦6
=
2 𝑦1∕2
𝐷24 =
4
𝐷96 =
𝐷104 =
1 𝑐5 𝑦6
2 𝑦1∕2
4
1
2
1∕2
𝑐3 𝑦4
1∕2
𝑦6
𝐷14 =
1
2
1∕2
𝑐4 𝑦3
1∕2
𝑦4
𝐷15 = 𝑐6 𝑦4
1 𝑐6 𝑦5
2 𝑦1∕2
4
1∕2
The matrix elements 𝐴𝑖𝑗 are
𝐴11 = 1 + 𝐷103
𝐴12 = 1 + 𝐷14 + 𝐷24 + 𝐷84 + 𝐷104 + 𝐷94
𝐴13 = 1 + 𝐷15
𝐴14 = 1 + 𝐷26 + 𝐷76 + 𝐷96
𝐴21 = 0
𝐴22 = 2𝐷24 + 𝐷94 − 𝑑1 𝐷14
𝐴23 = −𝑑1 (1 + 𝐷15 )
𝐴24 = 2𝐷26 + 2 + 𝐷76 + 𝐷96
𝐴31 = 𝐷103
𝐴32 = 2 + (2 − 𝑑2 )𝐷14 + 𝐷24 + 𝐷84 + 𝐷104 + 𝐷94
𝐴33 = 1 − 𝑑2 + (2 − 𝑑2 )𝐷15
𝐴34 = 𝐷26 + 𝐷96
𝐴41 = 2 + 𝐷103
𝐴42 = 𝐷104 + 𝑑3 𝐷14
𝐴43 = −𝑑3 (1 + 𝐷15 )
𝐴44 = 0
Evaluating the three thermodynamic derivatives 𝑐𝑝 = (𝜕ℎ∕𝜕𝑇 )𝑃 , (𝜕𝑣∕𝜕𝑇 )𝑃 , and
(𝜕𝑣∕𝜕𝑃 )𝑇 requires the change in the mole fractions 𝑦𝑖 due to changes in temperature
and pressure. The four independent mole fraction (𝑦3 , 𝑦4 , 𝑦5 , 𝑦6 ) derivatives with respect
to temperature are
𝜕𝑦3 𝜕𝑦4 𝜕𝑦5
,
,
,
𝜕𝑇 𝜕𝑇 𝜕𝑇
and
𝜕𝑦6
𝜕𝑇
and the remaining derivatives are expressed in terms of this independent set.
𝜕𝑦7
𝜕𝑇
𝜕𝑦8
𝜕𝑇
𝜕𝑦9
𝜕𝑇
𝜕𝑦10
𝜕𝑇
𝜕𝑦2
𝜕𝑇
𝜕𝑦1
𝜕𝑇
𝜕𝑦
+ 𝐷76 6
𝜕𝑇
𝜕𝑇
𝜕𝑦
1∕2 𝜕𝑐2
= 𝑦4
+ 𝐷84 4
𝜕𝑇
𝜕𝑇
𝜕𝑦
𝜕𝑦
1∕2 1∕2 𝜕𝑐3
= 𝑦4 𝑦6
+ 𝐷94 4 + 𝐷96 6
𝜕𝑇
𝜕𝑇
𝜕𝑇
1∕2 1∕2 𝜕𝑐4
= 𝑦4 𝑦3
𝜕𝑇
𝜕𝑦
𝜕𝑦
1∕2 𝜕𝑐
= 𝑦4 𝑦6 5 + 𝐷24 4 + 𝐷26 6
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑦
𝜕𝑦
1∕2 𝜕𝑐
= 𝑦4 𝑦5 6 + 𝐷14 4 + 𝐷15 5
𝜕𝑇
𝜕𝑇
𝜕𝑇
1∕2 𝜕𝑐1
= 𝑦6
(E.11)
Thermodynamic Analysis of Mixtures
407
These four independent derivatives are found by solution of the matrix equation that
results from differentiating Equation E.8 with respect to 𝑇 ,
𝜕𝑓𝑗
𝜕𝑇
+
𝜕𝑓𝑗 𝜕𝑦3 𝜕𝑓𝑗 𝜕𝑦4 𝜕𝑓𝑗 𝜕𝑦5 𝜕𝑓𝑗 𝜕𝑦6
+
+
+
=0
𝜕𝑦3 𝜕𝑇
𝜕𝑦4 𝜕𝑇
𝜕𝑦5 𝜕𝑇
𝜕𝑦6 𝜕𝑇
(E.12)
𝑗 = 1, 2, 3, 4
In matrix form,
[
[𝐀]
] [ ]
𝜕𝐲
𝜕𝐟
=0
+
𝜕𝑇
𝜕𝑇
(E.13)
where the matrix 𝐀 is identical to that used earlier to solve for the mole fractions. To
evaluate 𝜕𝑓𝑗 ∕𝜕𝑇 , define
𝑦1
𝑐6
𝑦2
𝑥2 =
𝑐5
𝑦
𝑥𝑘 = 𝑘
𝑐𝑘−6
𝑥1 =
(E.14)
where 𝑘 = 7, 8, 9, 10
Note that the terms 𝑥𝑘 are functions of 𝑦3 , 𝑦4 , 𝑦5 , and 𝑦6 only, so upon substitution
[
] of 𝑥𝑘
into Equations E.8 followed by differentiation with respect to 𝑇 yields the 𝜕𝑓 ∕𝜕𝑇 terms:
𝜕𝑐
𝜕𝑐
𝜕𝑐
𝜕𝑐
𝜕𝑐
𝜕𝑐
𝜕𝑓1
= 6 𝑥1 + 5 𝑥2 + 1 𝑥7 + 2 𝑥8 + 3 𝑥9 + 4 𝑥10
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑐6
𝜕𝑐3
𝜕𝑐5
𝜕𝑐1
𝜕𝑓2
=2
𝑥 +
𝑥 +
𝑥 − 𝑑1
𝑥
𝜕𝑇
𝜕𝑇 2 𝜕𝑇 7 𝜕𝑇 9
𝜕𝑇 1
𝜕𝑐
𝜕𝑐
𝜕𝑐
𝜕𝑓3
𝜕𝑐
𝜕𝑐
𝜕𝑐
= 2 6 𝑥1 + 5 𝑥2 + 2 𝑥8 + 3 𝑥9 + 4 𝑥10 − 𝑑2 6 𝑥1
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑐6
𝜕𝑐4
𝜕𝑓4
=
𝑥 − 𝑑3
𝑥
𝜕𝑇
𝜕𝑇 10
𝜕𝑇 1
From the definitions of 𝑐𝑖 , the 𝜕𝑐𝑖 ∕𝜕𝑇 terms are
(E.15)
𝜕𝑐1
𝑑𝐾4
𝜕𝑐4
1 𝑑𝐾1
=
=
1∕2
𝜕𝑇
𝑑𝑇
𝜕𝑇
𝑑𝑇
𝑃
𝑑𝐾5
𝜕𝑐
𝑑𝐾
𝜕𝑐2
1
5
2
=
= 𝑃 1∕2
(E.16)
1∕2
𝜕𝑇
𝑑𝑇
𝜕𝑇
𝑑𝑇
𝑃
𝑑𝐾6
𝜕𝑐6
𝑑𝐾3
𝜕𝑐3
=
= 𝑃 1∕2
𝜕𝑇
𝑑𝑇
𝜕𝑇
𝑑𝑇
As discussed in Chapter 3, the curve-fit equations for the equilibrium coefficients 𝐾𝑖
are of the form
𝐵
(E.17)
log10 𝐾𝑖 (𝑇 ) = 𝐴𝑖 ln(𝑇 ∕1000) + 𝑖 + 𝐶𝑖 + 𝐷𝑖 𝑇 + 𝐸𝑖 𝑇 2
𝑇
where 𝑇 is in Kelvin. Differentiation of Equation E.17 with respect to temperature
yields
[
]
𝑑𝐾𝑖
𝐴
𝐵
+ 𝐷 + 2𝐸𝑇
(E.18)
= 2.302585 𝐾𝑖
−
𝑑𝑇
𝑇
𝑇2
408
Thermodynamic Analysis of Mixtures
The evaluation of the mole fraction partial derivatives with respect to pressure parallels
the development of the temperature derivatives, with 𝜕∕𝜕𝑃 replacing 𝜕∕𝜕𝑇 . Note that since
neither 𝑐3 nor 𝑐4 depend on pressure, the terms containing 𝜕𝑐3 ∕𝜕𝑃 and 𝜕𝑐4 ∕𝜕𝑃 are zero.
The remaining derivatives are
𝜕𝑐1
1𝑐
=− 1
𝜕𝑃
2𝑃
𝜕𝑐5
1 𝑐5
=
𝜕𝑃
2𝑃
𝜕𝑐2
1𝑐
=− 2
𝜕𝑃
2𝑃
𝜕𝑐6
1𝑐
=− 6
𝜕𝑃
2𝑃
(E.19)
E.3 ISENTROPIC COMPRESSION/EXPANSION WITH KNOWN Δ𝑃
For an isentropic from pressure 𝑃1 to 𝑃2 , such as an exhaust blowdown, it follows that
𝑠2 = 𝑠1 . To find the unknown final temperature 𝑇2 , an iteration is required. We use an
efficient numerical root finding routine, the Newton--Raphson method. First, we define the
function 𝑓
𝑓 (𝑇2 ) = 𝑠2 − 𝑠(𝑇2 , 𝑃2 )
(E.20)
and then find the temperature 𝑇2 where the function 𝑓 (𝑇2 ) = 0, since 𝑃2 is known. Denote
the correct temperature as 𝑇2∗ and let 𝑇2(1) represent a first approximation to 𝑇2∗ . The function
𝑓 (𝑇2 ) can be expanded into a Taylor series about 𝑇2∗ . Let Δ𝑇 = 𝑇2∗ − 𝑇2(1) , then neglecting
higher order terms, one obtains
𝑓+
𝜕𝑓
Δ𝑇 = 0
𝜕𝑇
(E.21)
The derivative is evaluated at 𝑇2(1) and can be expressed by variables returned by
FARG and ECP.
The derivative required in Equation E.21 is
( )
𝑐p
𝜕𝑓
𝜕𝑠
=−
=−
𝜕𝑇
𝜕𝑇 𝑃
𝑇
(E.22)
Upon substitution into Equation E.21,
Δ𝑇 =
𝑓 𝑇2
𝑐𝑝2
(E.23)
Hence, an improved estimate of the temperature is given by
𝑇2(2)
=
𝑇2(1)
+
𝑓 𝑇2(1)
𝑐𝑝2
(E.24)
By substituting this second approximation into Equation E.23, an improved estimate of
Δ𝑇 is obtained. This procedure is repeated until Δ𝑇 is less than a specified tolerance. This
procedure converges rapidly, and the number of iterations is about 5--10 for the tolerances
used in the programs.
409
Thermodynamic Analysis of Mixtures
E.4 ISENTROPIC COMPRESSION/EXPANSION WITH KNOWN Δ𝑣
For an isentropic process such as compression or expansion from specific volume 𝑣1 to 𝑣2 ,
it also follows that 𝑠2 = 𝑠1 . We write two equations for the two unknowns 𝑇2 and 𝑃2 :
𝑠2 − 𝑠(𝑇2 , 𝑃2 ) = 0
(E.25)
𝑣2 − 𝑣(𝑇2 , 𝑃2 ) = 0
Using Newton--Raphson iteration, we define the following two functions 𝑓1 and 𝑓2 :
𝑓1 (𝑇2 , 𝑃2 ) = 𝑠2 − 𝑠(𝑇2 , 𝑃2 )
(E.26)
𝑓2 (𝑇2 , 𝑃2 ) = 𝑣2 − 𝑣(𝑇2 , 𝑃2 )
We denote the temperature 𝑇 and pressure 𝑃 where 𝑓1 and 𝑓2 equal zero as 𝑇2∗ , and
𝑃2∗ . Let 𝑇2(1) represent a first estimate of 𝑇2∗ , and 𝑃2(1) represent the first estimate of 𝑃2∗ .
The functions 𝑓1 and 𝑓2 can be expanded into a Taylor series about 𝑇2∗ and 𝑃2∗ . If we let
Δ𝑇 = 𝑇2∗ − 𝑇2(1) and Δ𝑃 = 𝑃2∗ − 𝑃2(1) , and neglect higher order terms, we obtain
𝑓1 +
𝜕𝑓1
𝜕𝑓
Δ𝑇 + 1 Δ𝑃 = 0
𝜕𝑇
𝜕𝑃
(E.27)
𝜕𝑓2
𝜕𝑓
Δ𝑇 + 2 Δ𝑃 = 0
𝜕𝑇
𝜕𝑃
The derivatives in Equation E.27 can be expressed in terms of the three derivatives 𝑐𝑝 ,
( )
( )
𝜕𝑣
𝜕𝑣
,
and
computed by the programs FARG and ECP:
𝜕𝑃
𝜕𝑇
𝑓2 +
𝑇
𝑃
( )
𝑐𝑝
𝜕𝑓1
𝜕𝑠
=−
=−
𝜕𝑇
𝜕𝑇 𝑃
𝑇
)
)
(
(
𝜕𝑓1
𝜕𝑣
𝜕𝑠
=
=−
𝜕𝑃
𝜕𝑃 𝑇
𝜕𝑇 𝑃
( )
𝜕𝑓2
𝜕𝑣
= − 𝜕𝑇
𝜕𝑇
𝑃
( )
𝜕𝑓2
𝜕𝑣
= − 𝜕𝑃
𝜕𝑃
𝑇
Upon substitution into Equation E.27, and solving for Δ𝑇 and Δ𝑃 ,
Δ𝑇 =
Δ𝑃 =
(𝜕𝑣∕𝜕𝑃 )𝑇 𝑓1 + (𝜕𝑣∕𝜕𝑇 )𝑃 𝑓2
𝑐𝑝 ∕𝑇 (𝜕𝑣∕𝜕𝑃 )𝑇 + (𝜕𝑣∕𝜕𝑇 )2𝑃
− (𝜕𝑣∕𝜕𝑇 )𝑃 𝑓1 + 𝑐𝑝 ∕𝑇 𝑓2
𝑐𝑝 ∕𝑇 (𝜕𝑣∕𝜕𝑃 )𝑇 + (𝜕𝑣∕𝜕𝑇 )2𝑃
(E.28)
(E.29)
(E.30)
(E.31)
An improved estimate of 𝑇2 and 𝑃2 is
𝑇2(2) = 𝑇2(1) + Δ𝑇
(E.32)
𝑃2(2) = 𝑃2(1) + Δ𝑃
By substituting these second approximations into Equation E.30, new values of Δ𝑇
and Δ𝑃 are obtained. This procedure is repeated until Δ𝑇 and Δ𝑃 are less than a specified
410
Thermodynamic Analysis of Mixtures
tolerance. This procedure converges rapidly, and the number of iterations is about 5--10 for
the tolerances used in the programs.
E.5 CONSTANT VOLUME COMBUSTION
If the combustion process from state 2 to 3 is assumed to be constant volume and adiabatic, it also follows that 𝑢3 = 𝑢2 , and the increase in 𝑇 and 𝑃 is due to the change in
chemical composition from an unburned fuel--air mixture to an equilibrium combustion
product mixture. Using the same numerical Newton--Raphson procedure as used in the
compression/expansion analyses, but with the constant thermodynamic parameters being 𝑢
and 𝑣 instead of 𝑠 and 𝑣, we first write two equations for the two unknowns 𝑇3 and 𝑃3 :
𝑢3 − 𝑢(𝑇3 , 𝑃3 ) = 0
𝑣3 − 𝑣(𝑇3 , 𝑃3 ) = 0
(E.33)
We then define the following two functions 𝑓1 and 𝑓2 :
𝑓1 (𝑇3 , 𝑃3 ) = 𝑢3 − 𝑢(𝑇3 , 𝑃3 )
(E.34)
𝑓2 (𝑇3 , 𝑃3 ) = 𝑣3 − 𝑣(𝑇3 , 𝑃3 )
To find where 𝑓1 and 𝑓2 are zero, we denote the correct temperature as 𝑇3∗ , the correct
pressure as 𝑃3∗ , let 𝑇3(1) represent a first estimate of 𝑇3∗ , and 𝑃3(1) represent the first estimate
of 𝑃3∗ . The functions 𝑓1 and 𝑓2 can be expanded into a Taylor series about 𝑇3∗ and 𝑃3∗ . If we
let Δ𝑇 = 𝑇3∗ − 𝑇3(1) and Δ𝑃 = 𝑃3∗ − 𝑃3(1) , then neglecting higher order terms, one obtains
Equation E.27.
The derivatives in Equation E.27 can be expressed in terms of the three derivatives 𝑐𝑝 ,
( )
( )
𝜕𝑣
𝜕𝑣
,
and
computed by FARG and ECP:
𝜕𝑇
𝜕𝑃
𝑃
𝑇
( )
𝜕𝑓1 ( 𝜕𝑢 )
𝜕𝑣
= 𝑐𝑝 − 𝑃
=
𝜕𝑇
𝜕𝑇 𝑃
𝜕𝑇 𝑃
( )
( )
( )
𝜕𝑓1
𝜕𝑣
𝜕𝑣
𝜕𝑢
= −𝑇
−𝑃
=−
𝜕𝑃
𝜕𝑃 𝑇
𝜕𝑇 𝑃
𝜕𝑃 𝑇
( )
𝜕𝑓2
𝜕𝑣
=−
𝜕𝑇
𝜕𝑇 𝑃
( )
𝜕𝑓2
𝜕𝑣
=−
𝜕𝑃
𝜕𝑃 𝑇
(E.35)
Upon substitution into Equation E.35, and solving for Δ𝑇 and Δ𝑃 ,
Δ𝑇 =
]
[
(−𝜕𝑣∕𝜕𝑃 )𝑇 𝑓1 − 𝑃 (𝜕𝑣∕𝜕𝑃 )𝑇 + 𝑇 (𝜕𝑣∕𝜕𝑇 )𝑃 𝑓2
Δ𝑃 =
𝑐p (𝜕𝑣∕𝜕𝑃 )𝑇 + 𝑇 (𝜕𝑣∕𝜕𝑇 )2𝑃
]
[
(𝜕𝑣∕𝜕𝑇 )𝑃 𝑓1 + 𝑃 (𝜕𝑣∕𝜕𝑇 )𝑃 − 𝑐p 𝑓2
𝑐p (𝜕𝑣∕𝜕𝑃 )𝑇 + 𝑇 (𝜕𝑣∕𝜕𝑇 )2𝑃
(E.36)
(E.37)
Thermodynamic Analysis of Mixtures
411
An improved estimate of 𝑇3 and 𝑃3 is
𝑇3(2) = 𝑇3(1) + Δ𝑇
(E.38)
𝑃3(2) = 𝑃3(1) + Δ𝑃
By substituting these second approximations into Equations E.36 and E.37, new values
of Δ𝑇 and Δ𝑃 are obtained. This procedure is repeated until Δ𝑇 and Δ𝑃 are less than a
specified tolerance. This procedure converges rapidly, and the number of iterations is about
5--10 for the tolerances used in the programs.
E.6 QUALITY OF EXHAUST PRODUCTS
The exhaust products can include water both in the vapor and in the liquid states. To obtain
the equilibrium quality 𝜒eq of the exhaust products, we assume the partial pressure of the
water vapor is equal to the saturation pressure corresponding to the mixture temperature,
that is,
𝑛H2 O,g
𝑛g
=
𝑃sat
𝑃
(E.39)
where 𝑛g is the number of moles of exhaust products in the gas phase. The total number of
moles of exhaust products, 𝑛 is the sum of the gas phase and liquid phase moles,
𝑛 = 𝑛g + 𝑛H2 O,f
(E.40)
𝑛H2 O = 𝑛H2 O,f + 𝑛H2 O,g
(E.41)
and since
we have
(
𝑛H2 O,g = (𝑛 − 𝑛H2 O )
)−1
𝑃
−1
𝑃sat
(E.42)
Therefore, for an equilibrium mixture of products,
𝜒eq =
𝑛H2 O,g
𝑛H2 O
(
=
1
𝑦H2 O
)(
−1
)−1
𝑃
−1
𝑃sat
(E.43)
If the combustion is stoichiometric, using the chemical formula of the fuel represented by
C𝑎 H𝑏 O𝑐 N𝑑 , the equilibrium quality 𝜒eq can be expressed as
𝜒eq =
(
9.52𝑎 + 1.88𝑏 − 3.76𝑐 + 𝑑
𝑏
)(
)−1
𝑃
−1
𝑃sat
(E.44)
412
Thermodynamic Analysis of Mixtures
EXAMPLE E.1:
Quality of Exhaust Products
Compute the quality 𝜒eq of the water in exhaust products for the stoichiometric combustion
of cetane. Assume the exhaust products are at standard reference conditions 𝑃 = 𝑃o = 1
bar and 𝑇o = 298.15 K.
SOLUTION The stoichiometric combustion equation per mole of cetane (𝑎 = 16, 𝑏 = 34, 𝑐 = 0,
and 𝑑 = 0) is
C16 H34 + 24.5(O2 + 3.76N2 ) → 16CO2 + 17H2 O + 92N2
At 298.15 K, the saturation pressure 𝑃sat of water vapor is 3.17 kPa, so upon substitution
in Equation E.43, we have
(
𝜒eq =
9.52(16) + 1.88(34) − 0 + 0
34
)(
101.3
−1
3.169
)−1
= 0.208
E.7 REFERENCES
Bridgman, P. (1914), ‘‘A Complete Collection of Thermodynamic Formulas,’’ Phys. Rev., Vol. 3, pp.
273--281.
Appendix
F
Computer Programs
The MATLABⓇ programs contained in this section are listed below. The programs
are also available online at the John Wiley web site and at www.engr.colostate.edu/
allan/engines.html.
Volume.m Computes and plots the exact and approximate cylinder volume versus crank angle.
Velocity.m Computes and plots the piston velocity versus crank angle.
BurnFraction.m Computes and plots the burn fraction versus crank angle.
FiniteHeatRelease.m
profile.
Computes pressure profile, work, efficiency, and imep for a given burn
FiniteHeatMassLoss.m Computes pressure profile, work, efficiency, and imep, including heat and
mass loss.
FourStrokeOtto.m Computes volumetric and thermal efficiency, residual fraction, and states for
four-stroke Otto engine.
RunFarg.m Input/output file for fuel--air--residual gas mixture program farg.m.
farg.m Computes mole fractions and thermodynamic properties of a fuel--air--residual gas mixture.
fuel.m Fuel property file.
RunEcp.m Input/output file for program ecp.m.
ecp.m Computes mole fractions and thermodynamic properties for equilibrium combustion of
fuel--air mixture.
AdiabaticFlameTemp.m
mixture.
Computes constant pressure adiabatic flame temperature of a fuel--air
OttoFuel.m Computes states, work, imep, efficiency of an Otto fuel--air cycle.
FourStrokeFuelAir.m
cycle.
Computes states, work, imep, efficiency for a four-stroke Otto fuel--air
Homogeneous.m Computes states, work, imep, efficiency and NO of a two--zone heat release
fuel--air cycle.
Friction.m Computes component and overall friction mean effective pressure.
WoschniHeatTransfer.m Computes pressure profile, work, efficiency, and imep, including
Woschni heat and mass loss.
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
413
414
Computer Programs
F.1 VOLUME.M
function [ ]=Volume( )
clear();
r = 10; % compression ratio
s = 80;
% stroke (cm)
len= 120; %connecting rod length (cm)
ep=s/(2*len);
theta=-180:1:180; %crankangle theta vector
ys1=(1-cosd(theta))/2; %approx y/s
ys2= ys1+ (1-(1- epˆ2*sind(theta).ˆ2).ˆ(1/2))/(2*ep); %exact y/s
vol1 = 1+(r-1)*ys1; %approx volume
vol2= 1+(r-1)*ys2;
% exact volume
%plot results
plot(theta,vol1,’--’,theta,vol2,’-’,’linewidth’,2);
set(gca,’Xlim’,[-180 180],’Ylim’,[0 r],’fontsize’,18,’linewidth’,2);
xlabel(’Crank Angle (deg)’,’fontsize’, 18);
ylabel(’Dim. Cylinder Volume’,’fontsize’, 18);
legend(’Approx. Volume’, ’Exact Volume’,’Location’, ’North’);
end
F.2 VELOCITY.M
function [ ]=Velocity( )
clear();
N=2000; %rev/min
s = 0.080;
% stroke (m)
len= 0.120; %connecting rod length (m)
ep=s/(2*len);
theta=0:1:180; %crankangle theta vector
term1=pi/2*sind(theta);
term2= (1+(ep*cosd(theta))./(1 - epˆ2*sind(theta).ˆ2).ˆ(1/2)); %exact y/s
Upbar=term1.*term2;
Up=Upbar*2*N*s/60;
%plot results
plot(theta,Up,’linewidth’,2);
set(gca,’Xlim’,[0 180],’fontsize’,18,’linewidth’,2);
xlabel(’Crank Angle (deg)’,’fontsize’, 18);
ylabel(’ Piston Velocity (m/s)’,’fontsize’, 18);
end
F.3 BURNFRACTION.M
function [ ]=BurnFraction( )
% this program computes and plots the cumulative burn fraction
% and the instantanous burnrate
clear();
Computer Programs
a = 5; % Weibe efficiency factor
n = 3;
% Weibe form factor
thetas = -20; % start of combustion
thetad = 60;
% duration of combustion
theta=linspace(thetas,thetas+thetad,100); %crankangle theta vector
dum=(theta-thetas)/thetad; % theta diference vector
temp=-a*dum.ˆn;
xb=1.-exp(temp); %burn fraction
dxb=n*a*(1-xb).*dum.ˆ(n-1); %element by element vector multiplication
%plot results
plot(theta,xb,’b’,’linewidth’,2);
set(gca, ’fontsize’, 18,’linewidth’,2);
xlabel(’Crank Angle (deg)’,’fontsize’, 18);
ylabel(’Cumulative Burn Fraction’,’fontsize’, 18);
figure();
plot(theta,dxb,’b’,’linewidth’,2);
set(gca, ’fontsize’, 18,’linewidth’,2);
xlabel(’Crank Angle (deg)’,’fontsize’, 18);
ylabel(’Burn Rate (1/deg)’,’fontsize’, 18);
end
F.4 FINITEHEATRELEASE.M
function [ ]=FiniteHeatRelease()
% Gas cycle heat release code for two engines
% engine parameters
clear();
thetas(1,1)= -10; % Engine1 start of heat release (deg)
thetas(2,1)= -10;
% Engine2 start of heat release (deg)
thetad(1,1) = 40; % Engine1 duration of heat release (deg)
thetad(2,1) = 10; % Engine2 duration of heat release (deg)
r=10;
%compression ratio
gamma= 1.4; %gas const
q= 34.8;
a= 5;
% dimensionless total heat release Qin/P1V1
%weibe parameter a
n= 3; %weibe exponent n
step=1;
% crankangle interval for calculation/plot
NN=360/step; % number of data points
% initialize the results data structure
save.theta=zeros(NN,1); % crankangle
save.vol=zeros(NN,1);
% volume
save.press=zeros(NN,2); % pressure
save.work=zeros(NN,2);
% work
pinit(1) = 1; % Engine 1 initial dimensionless pressure P/P1
pinit(2) = 1; % Engine 2 initial dimensionless pressure P/P1
% for loop for engine1 and engine2
for j=1:2
415
416
Computer Programs
theta = -180;
%initial crankangle
thetae = theta + step; %final crankangle in step
fy(1) = pinit(j); % assign initial pressure to working vector
fy(2) =0.;
% reset work vector
% for loop for pressure and work calculation
for i=1:NN,
[fy, vol] = integrate(theta,thetae,fy);
% reset to next interval
theta = thetae;
thetae = theta+step;
% copy results to output vectors
save.theta(i)=theta;
save.vol(i)=vol;
save.press(i,j)=fy(1);
save.work(i,j)=fy(2);
end %end of pressure and work iteration loop
end %end of engine iteration loop
[pmax1, id_max1] = max(save.press(:,1)); %Engine 1 max pressure
[pmax2, id_max2] = max(save.press(:,2)); %Engine 2 max pressure
thmax1=save.theta(id_max1);%Engine 1 crank angle
thmax2=save.theta(id_max2);%Engine 2 crank angle
w1=save.work(NN,1);
w2=save.work(NN,2);
eta1= w1/q; % thermal efficiency
eta2= w2/q;
imep1 = eta1*q*(r/(r -1)); %imep
imep2 = eta2*q*(r/(r -1));
eta_rat1 = eta1/(1-rˆ(1-gamma));
eta_rat2 = eta2/(1-rˆ(1-gamma));
% output overall results
fprintf(’
fprintf(’ Theta_start
fprintf(’ Theta_dur
fprintf(’ P_max/P_1
fprintf(’ Theta_max
Engine 1
%5.2f
%5.2f
fprintf(’ Efficiency
%7.1f
fprintf(’ Eff. Ratio
fprintf(’ Imep/P1
%5.2f \n’, pmax1, pmax2);
%7.1f \n’,thmax1,thmax2);
%7.2f
%5.3f
%5.3f
%5.2f
%7.2f \n’, w1,w2);
%5.3f
%5.3f
%5.2f
\n’, eta1, eta2);
\n’, eta_rat1, eta_rat2);
\n’, imep1, imep2);
%plot results
plot(save.theta,save.press(:,1),’-’,...
save.theta,save.press(:,2),’--’,’linewidth’,2 )
set(gca, ’fontsize’, 18,’linewidth’,2);
legend(’Engine 1’, ’Engine 2’,’Location’,’NorthWest’)
xlabel(’Theta (deg)’,’fontsize’, 18)
ylabel(’Pressure (bar)’,’fontsize’, 18)
print -deps2 heatrelpressure
\n’);
%5.2f \n’, thetad(1,1), thetad(2,1));
%5.2f
fprintf(’ Net Work/P1V1
Engine 2
%5.2f \n’, thetas(1,1), thetas(2,1));
Computer Programs
figure( );
plot(save.theta,save.work(:,1),’-’, ...
save.theta,save.work(:,2),’--’, ’linewidth’,2)
set(gca, ’fontsize’, 18,’linewidth’,2);
legend(’Engine 1’, ’Engine 2’,’Location’,’NorthWest’)
xlabel(’Theta (deg)’,’fontsize’, 18)
ylabel(’Work’,’fontsize’, 18)
function[fy,vol] = integrate(theta,thetae,fy)
%ode23 integration of the pressure differential equation
%from theta to thetae with current values of fy as initial conditions
[tt, yy] = ode23(@rates, [theta thetae], fy);
for k=1:2
fy(k) = yy(length(tt),k); %put last element of yy into fy vector
end
%pressure differential equation
function [yprime] = rates(theta,fy)
vol=(1.+ (r -1)/2.*(1-cosd(theta)))/r;
dvol=(r - 1)/2.*sind(theta)/r*pi/180.; %dvol/dtheta
dx=0.; %set heat release to zero
if(theta > thetas(j)) % then heat release dx
> 0
dum1=(theta -thetas(j))/thetad(j);
x=1.- exp(-(a*dum1ˆn));
dx=(1-x)*a*n*dum1ˆ(n-1)/thetad(j); %dx/dthetha
end
term1= -gamma*fy(1)*dvol/vol;
term2= (gamma-1)*q*dx/vol;
yprime(1,1)= term1 + term2;
yprime(2,1)= fy(1)*dvol;
end %end of function rates
end
%end of function integrate2
end % heat_release_weibe2
F.5 FINITEHEATMASSLOSS.M
function [ ] = FiniteHeatMassLoss( )
% Gas cycle heat release code with and w/o heat transfer
% data structure for engine parameters
clear();
thetas = -20; % start of heat release (deg)
thetad = 40; % duration of heat release (deg)
r =10;
% compression ratio
gamma = 1.4; % gas const
Q = 20.;
% dimensionless total heat release
h = 0.2;
% dimensionless ht coefficient
tw = 1.2;
% dimensionless cylinder wall temp
beta = 1.5; % dimensionless volume
a = 5;
% weibe parameter a
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Computer Programs
n = 3;
% weibe exponent n
omega =200.; % engine speed rad/s
c = 0.8;
step=1;
% mass loss coeff
% crankangle interval for calculation/plot
NN=360/step; % number of data points
theta = -180; % initial crankangle
thetae = theta + step; % final crankangle in step
% initialize results data structure
save.theta=zeros(NN,1);
save.vol=zeros(NN,1);
% volume
save.press=zeros(NN,1); % pressure
save.work=zeros(NN,1);
% work
save.heatloss=zeros(NN,1); % heat loss
save.mass=zeros(NN,1);
% mass left
fy=zeros(4,1); % vector for pressure, work, heat and mass loss
fy(1) = 1; % initial pressure (bar)
fy(4) = 1; % initial mass (-)
%for loop for pressure and work calculation
for i=1:NN,
[fy, vol] = integrate_ht(theta,thetae,fy);
% print values
% fprintf(’%7.1f
%7.2f
%7.2f
%7.2f \n’, theta,vol,fy(1),fy(2),fy(3));
% reset to next interval
theta = thetae;
thetae = theta+step;
save.theta(i)=theta; % put results in output vectors
save.vol(i)=vol;
save.press(i)=fy(1);
save.work(i)=fy(2);
save.heatloss(i)=fy(3);
save.mass(i)=fy(4);
end % end of pressure and work for loop
[pmax, id_max] = max(save.press(:,1)); % find max pressure
thmax=save.theta(id_max);
% and crank angle
ptdc=save.press(NN/2)/pmax;
w=save.work(NN,1);
% w is cumulative work
massloss =1- save.mass(NN,1);
eta=w/Q;
% thermal efficiency
imep = eta*Q*(r/(r -1)); %imep/P1V1
eta_rat = eta/(1-rˆ(1-gamma));
% output overall results
fprintf(’ Weibe Heat Release with Heat and Mass Loss
fprintf(’ Theta_start =
%5.2f
\n’, thetas);
fprintf(’ Theta_dur =
%5.2f
\n’, thetad);
fprintf(’ P_max/P1 =
%5.2f
\n’, pmax);
fprintf(’ Theta @P_max =
fprintf(’ P_tdc/P_max =
%7.1f
\n’,thmax);
%5.2f
\n’, ptdc);
\n’);
Computer Programs
fprintf(’ Net Work/P1V1 =
%7.2f
\n’, w);
fprintf(’ Heat Loss/P1V1 =
%7.2f
\n’, save.heatloss(NN,1));
fprintf(’ Mass Loss/m =
%7.3f
fprintf(’ Efficiency =
%5.3f
\n’,massloss );
\n’, eta);
fprintf(’ Eff./Eff. Otto =
%5.3f
\n’, eta_rat);
fprintf(’ Imep/P1 =
%5.2f
\n’, imep);
%plot results
plot(save.theta,save.press,’-’,’linewidth’,2 )
set(gca, ’fontsize’, 18,’linewidth’,1.5);
xlabel(’Crank Angle \theta (deg)’,’fontsize’, 18)
ylabel(’Pressure P (bar)’,’fontsize’, 18)
figure();
plot(save.theta,save.work,’-’,save.theta,save.heatloss,’--’,’linewidth’,2 )
set(gca, ’Xlim’,[-180 180],’fontsize’, 18,’linewidth’,1.5);
hleg1=legend(’Work’, ’Heat Loss’,’Location’,’NorthWest’)
set(hleg1,’Box’, ’off’)
xlabel(’Crank Angle \theta (deg)’,’fontsize’, 18)
ylabel(’Cumulative Work and Heat Loss’,’fontsize’, 18)
function[fy,vol] = integrate_ht(theta,thetae,fy)
%
ode23 integration of the pressure differential equation
%
from theta to thetae with current values of fy as initial conditions
[tt, yy] = ode23(@rates, [theta thetae], fy);
% put last element of yy into fy vector
for j=1:4
fy(j) = yy(length(tt),j);
end
% pressure differential equation
function [yprime] = rates(theta,fy)
vol=(1.+ (r -1)/2.*(1-cosd(theta)))/r;
dvol=(r - 1)/2.*sind(theta)/r*pi/180.; %dvol/dtheta
dx=0.;
if(theta>thetas) % heat release >0
dum1=(theta -thetas)/thetad;
x=1-exp(-(a*dum1ˆn));
dx=(1-x)*a*n*dum1ˆ(n-1)/thetad; %dx/dthetha
end
term1= -gamma*fy(1)*dvol/vol;
term3= h*(1. + beta*vol)*(fy(1)*vol/fy(4) - tw)*pi/180.;
term2= (gamma-1)/vol*(Q*dx - term3);
yprime(1,1)= term1 + term2 - gamma*c/omega*fy(1)*pi/180;
yprime(2,1)= fy(1)*dvol;
yprime(3,1)= term3;
yprime(4,1)= -c*fy(4)/omega*pi/180;
end %end of function rates
end % end of function integrate_ht
end % end of function HeatReleaseHeatTransfer
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F.6 FOURSTROKEOTTO.M
% Four stroke Otto cycle model
% input parameters
Ti = 300; % inlet temperature, K
Pi = 50; % inlet pressure, kPa
Pe = 100; % exhaust pressure, kPa
r = 10; % compression ratio
qin = 2500; % heat input, kJ/kg (mixture)
gamma = 1.3; % ideal gas specific heat ratio
R = 0.287; % gas constant kJ/kg K
cv= R/(gamma-1); %const vol specific heat, kJ/kg K
f=0.05;% guess value of residual fraction f
Tr = 1000; % guess value of exhaust temp, K
tol=0.001; % tolerance for convergence
err = 2*tol; %error initialization
gam=(gamma -1)/gamma;
while (err > tol) %while loop for cycle calc
%intake stroke
T1=(1-f)*Ti + f*(1 - (1- Pi/Pe)*gam)*Te;
P1=Pi;
%isentropic compression
P2=P1*rˆgamma;
T2=T1*rˆ(gamma-1);
%const v heat addition
T3=T2 + qin*(1-f)/cv;
P3=P2*(T3/T2);
%isentropic expansion
P4=P3*(1/r)ˆgamma;
T4=T3*(1/r)ˆ(gamma-1);
%isentropic blowdown
P5=Pe;
T5=T4*(P4/Pe)ˆ(-gam);
%const p exhaust stroke
Te=T5;
fnew=(1/r)*(Pe/P4)ˆ(1/gamma); %new residual fraction
err=abs(fnew-f)/fnew;
f=fnew;
end
%cycle parameters
eta= 1 - 1/rˆ(gamma-1);
imep = P1*(qin*(1-f)/(R*T1))/(1-(1/r))*eta;
pmep=Pe-Pi;
etanet= eta*(1-pmep/imep);
imepnet= (imep-pmep)/100.;
voleff=1-(Pe/Pi -1)/(gamma*(r-1));
%output calcs
fprintf(’ \nFour Stroke Otto Cycle
\n’)
Computer Programs
fprintf(’State
1
2
fprintf(’Pressure (kPa):
%6.1f
%6.1f
fprintf(’Temperature (K):
%6.1f
%6.1f
fprintf(’Ideal Thermal Eff.= %6.3f
3
4 \n’);
%6.1f %6.1f \n’,P1,P2,P3,P4);
%6.1f
%6.1f \n’,T1,T2,T3,T4);
Net Thermal Eff.=
%6.3f \n’, ...
eta, etanet);
fprintf(’Exhaust Temp. (K)=
%6.1f
Volumetric Eff.=
%6.3f
Net Imep (bar)=
%6.2f \n’, ...
Te, voleff);
fprintf(’Residual Fraction
%6.2f \n’,...
f, imepnet);
F.7 RUNFARG.M
%Input-Output program for running farg.m
clear;
T = 500; % enter temperature (K) input
P = 100.;
% enter pressure (kPa) input
phi = 0.8;
% enter equivalence ratio input
f=0.1; %residual fraction input
fuel_id = 1;
% fuel_id
-
421
1=Methane, 2=Gasoline, 3=Diesel, 4=Methanol, 5=Nitromethane
% call farg function
[Y,h,u,s,v,R,Cp,MW,dvdT,dvdP,dMWdT,dMWdP] = farg( T, P, phi, f, fuel_id );
%echo input
fprintf(’ \n Fuel Air Residual Gas \n’ );
fprintf(’ Pressure (kPa) =
%6.1f \n’, P );
fprintf(’ Temperature (K) =
%6.1f \n’, T); ...
fprintf(’ Fuel Air Equivalence ratio = % 3.1f \n’, phi);
fprintf(’ Residual Fraction = \t% 3.1f \n ’, f);
%print output mole fractions and properties
fprintf(’ \n Mole Fractions \n’ );
fprintf(’ CO2 = \t %6.4f \n’, Y(1) );
fprintf(’ H2O = \t %6.4f \n’, Y(2) );
fprintf(’ N2 = \t %6.4f \n’, Y(3) );
fprintf(’ O2 = \t %6.4f \n’, Y(4) );
fprintf(’ CO = \t %6.4f \n’, Y(5) );
fprintf(’ H2 = \t %6.4f \n’, Y(6) );
fprintf(’ H = \t %6.4f \n’, Y(7) );
fprintf(’ O = \t %6.4f \n’, Y(8) );
fprintf(’ OH = \t %6.4f \n’, Y(9) );
fprintf(’ NO = \t %6.4f \n’, Y(10) );
fprintf(’ \n Mixture Properties \n’ );
fprintf(’ h(kJ/kg) = \t %6.1f \n’, h );
fprintf(’ u(kJ/kg) = \t %6.1f \n’, u );
fprintf(’ s (kJ/Kg K) = \t %6.3f \n’, s );
fprintf(’ v (m3/kg) = \t %6.3f \n’, v );
fprintf(’ Cp (kJ/Kg K) =\t %6.3f \n’, Cp );
fprintf(’ Molecular Mass = %5.2f \n’, MW );
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Computer Programs
fprintf(’ dvdt = %8.2e \n’, dvdT );
fprintf(’ dvdp = %8.2e \n’, dvdP );
F.8 FARG.M
function [Y,h,u,s,v,R,Cp,MW,dvdT,dvdP]=farg(T,P,phi,f,fuel_id)
% Subroutine for Fuel Air Residual Gas
%
% inputs:
%
T - temperature (K)
%
P - pressure (kPa)
[ 300 --> 1000 K ]
%
phi - equivalence ratio
%
f - residual fraction
%
fuel_id - 1=Methane, 2=Gasoline, 3=Diesel, 4=Methanol, 5=Nitromethane
%
% outputs:
%
y - mole fraction of constituents
%
y(1)
: CO2
%
y(2)
: H2O
%
y(3)
: N2
%
y(4)
: O2
%
y(5)
: CO
%
y(6)
: H2
%
h
- specific enthalpy of mixture, kJ/kg
%
u
- specific internal energy of mixture, kJ/kg
%
s
- specific entropy of mixture, kJ/kgK
%
v
- specific volume of mixture, m3/kg
%
r
- specific ideal gas constant, kJ/kgK
%
cp - specific heat at constant pressure, kJ/kgK
%
mw - molecular weight of mixture, kg/kmol
%
dvdt - (dv/dT) at const P,
m3/kg per K
%
dvdp - (dv/dP) at const T,
m3/kg per kPa
% Get fuel composition information
[alpha,beta,gamma,delta,h_fuel,so_fuel,cp_fuel,m_fuel ]=fuel( fuel_id, T );
% Curve fit coefficients for thermodynamic properties
%
300 < T < 1000 K
% Cp/R = a1 + a2*T + a3*Tˆ2 + a4*Tˆ3 + a5*Tˆ4
% h/RT = a1 + a2/2*T + a3/3*Tˆ2 + a4/4*Tˆ3 + a5/5*Tˆ4 + a6/T
% so/R = a1*ln(T) + a2*T + a3/2*Tˆ2 + a4/3*Tˆ3 + a5/4*Tˆ4 + a7
A = [
[ 0.24007797e+1,
0.87350957e-2, -0.66070878e-5,
0.63274039e-15, -0.48377527e+5,
0.20021861e-8,
0.96951457e+1 ]; ... % CO2
[ 0.40701275e+1, -0.11084499e-2,
0.41521180e-5, -0.29637404e-8,
0.80702103e-12, -0.30279722e+5, -0.32270046
[ 0.36748261e+1, -0.12081500e-2,
-0.2257725e-12, -0.10611588e+4,
0.23580424e+1 ]; ... % N2
[ 0.36255985e+1, -0.18782184e-2,
0.21555993e-11, -0.10475226e+4,
]; ... % H2O
0.23240102e-5, -0.63217559e-9,
0.70554544e-5, -0.67635137e-8,
0.43052778e+1 ]; ... % O2
Computer Programs
[ 0.37100928e+1, -0.16190964e-2,
0.23953344e-12, -0.14356310e+5,
[ 0.30574451e+1,
0.36923594e-5, -0.20319674e-8,
0.2955535e+1
]; ... % CO
0.26765200e-2, -0.58099162e-5,
-0.1812273e-11, -0.98890474e+3, -0.22997056e+1 ] ];
0.55210391e-8,
% H2
% molar mass of constituents
%
CO2
Mi = [ 44.01,
H2O
N2
O2
CO
H2
18.02,
28.013,
32.00,
28.01,
2.016 ];
% Calculate stoichiometric molar air-fuel ratio
a_s = alpha + beta/4 - gamma/2;
% mole fraction of fuel, O2, N2
y_1 = 1 / (1 + 4.76*a_s/phi);
% mole fraction for one mole of reactant
y_fuel = y_1; % assuming 1 mole fuel
y_O2 = a_s/phi * y_1; % a_s/phi moles O2
y_N2 = a_s/phi*3.76 * y_1; % a_s/phi * 3.76 moles N2
% mass of fuel air mixture (M’’)
m_fa = y_fuel*m_fuel + y_O2*32.00 + y_N2*28.013;
% default case: no residual gas
Y = zeros(6,1);
m_r = 0; % mass of residual gas
y_r = 0; % mole fraction of residual gas in mixture
n = zeros(6,1);
dcdt = 0;
if ( phi <= 1 )
% lean combustion
n(1) = alpha;
n(2) = beta/2;
n(3) = delta/2 + 3.76*a_s/phi;
n(4) = a_s*(1/phi - 1);
else
% rich combustion
d1 = 2*a_s*(1-1/phi);
z = T/1000;
K = exp( 2.743 - 1.761/z - 1.611/zˆ2 + 0.2803/zˆ3 );
a1 = 1-K;
b1 = beta/2 + alpha*K - d1*(1-K);
c1 = -alpha*d1*K;
n(5) = (-b1 + sqrt(b1ˆ2 - 4*a1*c1))/(2*a1);
% Required derivatives for Cp calculation of mixture
% calculate dcdt = dn5/dK * dK/dT
dkdt = -K*(-1.761+z*(-3.222+z*.8409))/1000;
dn5dk =
-((alpha - n(5))*(n(5) + 2*a_s*(1/phi - 1)))/(beta/2 + n(5)
+ 2*a_s*(1/phi - 1));
dcdt = dn5dk * dkdt;
n(1) = alpha - n(5);
n(2) = beta/2 - d1 + n(5);
n(3) = delta/2 + 3.76*a_s/phi;
n(6) = d1 - n(5);
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Computer Programs
end
% total moles
N = sum(n);
% calculate mole fractions and mass of residual gas
m_r = 0;
for i=1:6,
Y(i) = n(i)/N;
m_r = m_r + Y(i)*Mi(i);
end
% compute residual mole fraction
y_r = 1/(1 + m_r/m_fa * (1/f-1));
% compute total mole fractions in mixture
for i=1:6,
Y(i) = Y(i)*y_r;
end
% fuel mole fraction based on all moles
y_fuel = y_fuel*(1 - y_r);
% include intake N2 and O2
Y(3) = Y(3) + y_N2*(1 - y_r);
Y(4) = Y(4) + y_O2*(1 - y_r);
% compute properties of mixture
h = h_fuel*y_fuel;
s = (so_fuel-log(max(y_fuel,1e-15)))*y_fuel;
Cp = cp_fuel*y_fuel;
MW = m_fuel*y_fuel;
% compute component properties according to curve fits
cpo = zeros(6,1);
ho = zeros(6,1);
so = zeros(6,1);
for i=1:6
cpo(i) = A(i,1) + A(i,2)*T + A(i,3)*Tˆ2 + A(i,4)*Tˆ3 + A(i,5)*Tˆ4;
ho(i) = A(i,1) + A(i,2)/2*T + A(i,3)/3*Tˆ2 + A(i,4)/4*Tˆ3
+A(i,5)/5*Tˆ4 + A(i,6)/T;
so(i) = A(i,1)*log(T) + A(i,2)*T + A(i,3)/2*Tˆ2 + A(i,4)/3*Tˆ3
+A(i,5)/4*Tˆ4 +A(i,7);
end
table = [-1,1,0,0,1,-1];
for i=1:6
if(Y(i)>1.e-25)
h = h + ho(i)*Y(i);
s = s + Y(i)*(so(i)-log(Y(i)));
Cp = Cp+cpo(i)*Y(i)+ho(i)*T*table(i)*dcdt*y_r/N;
MW = MW + Y(i)*Mi(i);
end
end
% compute thermodynamic properties
R = 8.31434/MW; % compute mixture gas constant
Computer Programs
h = R*T*h; % curve fit for h is h/rt
u = h-R*T;
v = R*T/P;
s = R*(-log(P/101.325)+s);
Cp = R*Cp; % curve fit for cp is cp/r
dvdT = v/T; % derivative of volume wrt temp
dvdP = -v/P; % derivative of volume wrt pres
F.9 FUEL.M
function [alpha,beta,gamma,delta,h,s,cp,mw,Fs,q ] = fuel( id, T )
% [ alpha, beta, gamma, delta, h, s, cp, mw, Fs, q ] = fuel( id, T )
%
%
Parameters
%
id
-
1=Methane, 2=Gasoline, 3=Diesel, 4=Methanol, 5=Nitromethane
%
T
-
Temperature (K) at which to eval 300<T<1000 K
%
Outputs
%
alpha - # carbon
%
beta
%
gamma - # oxygen
%
delta - # nitrogen
%
h
- specific enthalpy (kJ/kg)
%
s
- specific entropy (kJ/kgK)
%
cp
- specific heat (kJ/kgK)
%
mw
- molecular weight (kg/kmol)
%
Fs
- stoichiometric fuel-air ratio
%
q
- heat of combustion (kJ/kg)
- # hydrogen
% Curve fit coefficients for thermodynamic properties of selected fuels
%
a1
a2
FuelProps = [
8.873728
a3
a6
[ 1.971324, 7.871586e-3,
a7
-1.048592e-06, -9.930422e+3,
]; ... % Methane
[ 4.0652,
6.0977e-2,
1.545e+1
[ 7.971,
-1.8801e-05, -3.588e+4,
]; ... % Gasoline
1.1954e-01, -3.6858e-05, -1.9385e+4,
-1.7879
]; ... % Diesel
[ 1.779819, 1.262503e-02, -3.624890e-6,-2.525420e+4,
1.50884e+1
]; ... % Methanol
[ 1.412633,
2.0871e-02,
1.917126e+1 ] ];
-8.14213e-6, -1.02635e+4,
% Nitromethane
% Fuel chemical formula
%
C
H
O
%
alpha
beta
gamma
FuelInfo = [[ 1
N
delta
4
0
0 ]; ... % Methane
[ 7
17
0
0 ]; ... % Gasoline
[ 14.4
24.9
0
0 ]; ... % Diesel
[ 1
4
1
0 ]; ... % Methanol
[ 1
3
2
1 ]
];
% Nitromethane
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Computer Programs
% stoichiometric fuel-air ratio
FSv = [ 0.0584
0.06548
0.06907
0.1555
0.5924 ];
% available energy of combustion ac
ac = [ 52420
47870
45730
22680
12430 ];
% stoichiometric fuel-air ratio
Fs = FSv(id);
% available energy
q = ac(id);
% Get fuel composition
alpha = FuelInfo(id, 1);
beta = FuelInfo(id, 2);
gamma = FuelInfo(id, 3);
delta = FuelInfo(id, 4);
% compute fuel properties
ao = FuelProps(id, 1);
bo = FuelProps(id, 2);
co = FuelProps(id, 3);
do = FuelProps(id, 4);
eo = FuelProps(id, 5);
% compute thermodynamic properties
h = ao + bo/2*T +co/3*Tˆ2 +do/T;
s = ao*log(T) + bo*T +co/2*Tˆ2 + eo;
cp = ao + bo*T + co*Tˆ2;
% Calculate molecular weight of fuel
mw = 12.01*alpha + 1.008*beta + 16.00*gamma + 14.01*delta;
F.10 RUNECP.M
%Input-Output program for ecp.m
clear;
phi = 0.8;
% enter equivalence ratio input
T = 3000; % enter temperature (K) input
P = 5000.;
% enter pressure (kPa) input
fuel_id =2;
%
fuel_id - 1=Methane, 2=Gasoline, 3=Diesel, 4=Methanol, 5=Nitromethane
% call ecp function
[ierr, Y, h, u, s, v, R, Cp, MW, dvdT, dvdP] = ecp( T, P, phi, fuel_id );
%echo input
fprintf(’ \n Equilibrium Combustion Solver \n’ );
fprintf(’ Pressure (kPa) = \t \t %6.1f \n’, P );
fprintf(’ Temperature (K) = \t \t %6.1f \n’, T); ...
fprintf(’ Fuel Air Equivalence ratio = \t% 3.1f \n ’, phi);
%print output mole fractions and properties
fprintf(’ \n Mole Fractions \n’ );
fprintf(’ CO2 = \t %6.4f \n’, Y(1) );
fprintf(’ H2O = \t %6.4f \n’, Y(2) );
fprintf(’ N2 = \t %6.4f \n’, Y(3) );
Computer Programs
fprintf(’ O2 = \t %6.4f \n’, Y(4) );
fprintf(’ CO = \t %6.4f \n’, Y(5) );
fprintf(’ H2 = \t %6.4f \n’, Y(6) );
fprintf(’ H = \t %6.4f \n’, Y(7) );
fprintf(’ O = \t %6.4f \n’, Y(8) );
fprintf(’ OH = \t %6.4f \n’, Y(9) );
fprintf(’ NO = \t %6.4f \n’, Y(10) );
fprintf(’ \n Mixture Properties \n’ );
fprintf(’ h(kJ/kg) = \t %6.1f \n’, h );
fprintf(’ u(kJ/kg) = \t %6.1f \n’, u );
fprintf(’ s (kJ/Kg K) = \t %6.3f \n’, s );
fprintf(’ v (m3/kg) = \t %6.3f \n’, v );
fprintf(’ cp (kJ/Kg K) =\t %6.3f \n’, Cp );
fprintf(’ Molecular Mass = %5.2f \n’, MW );
fprintf(’ dvdt = %8.2e \n’, dvdT );
fprintf(’ dvdp = %8.2e \n’, dvdP );
F.11 ECP.M
function [ierr,Y,h,u,s,v,R,Cp,MW,dvdT,dvdP]=ecp(T,P,phi,ifuel )
% Subroutine for Equilibrium Combustion Products
%
% inputs:
%
T - temperature (K)
[ 600 --> 3500 ]
%
P - pressure (kPa)
[ 20 --> 30000 ]
%
phi - equivalence ratio
[ 0.01 --> ˜3 ]
%
ifuel - 1=Methane, 2=Gasoline, 3=Diesel, 4=Methanol, 5=Nitromethane
%
% outputs:
%
ierr - Error codes:
%
0 = success
%
1 = singular matrix
%
2 = maximal pivot error in gaussian elimination
%
3 = no solution in maximum number of iterations
%
4 = result failed consistency check sum(Y)=1
%
5 = failure to obtain initial guess for oxygen concentration
%
6 = negative oxygen concentration in initial guess calculation
%
7 = maximum iterations reached in initial guess solution
%
%
8 = temperature out of range
9 = pressure out of range
% 10 = equivalence ratio too lean
%11 = equivalence ratio too rich, solid carbon will be formed
%
%
y - mole fraction of constituents
%
y(1)
: CO2
%
y(2)
: H2O
%
y(3)
: N2
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Computer Programs
%
y(4)
: O2
%
y(5)
: CO
%
y(6)
: H2
%
y(7)
: H
%
y(8)
: O
%
y(9)
: OH
%
y(10) : NO
%
h
- specific enthalpy of mixture, kJ/kg
%
u
- specific internal energy of mixture, kJ/kg
%
s
- specific entropy of mixture, kJ/kgK
%
v
- specific volume of mixture, m3/kg
%
R
- specific ideal gas constant, kJ/kgK
%
Cp - specific heat at constant pressure, kJ/kgK
%
MW - molecular weight of mixture, kg/kmol
%
dvdt - (dv/dT) at const P,
m3/kg per K
%
dvdp - (dv/dP) at const T,
m3/kg per kPa
% initialize outputs
Y = zeros(10,1);
h = 0;
u = 0;
s = 0;
v = 0;
R = 0;
Cp = 0;
MW = 0;
dvdT = 0;
dvdP = 0;
% solution parameters
prec = 1e-3;
MaxIter = 20;
% square root of pressure (used many times below)
PATM = P/101.325;
sqp = sqrt(PATM);
if ( T < 600 || T > 3500 )
ierr = 8;
return;
end
if ( P < 20 || P > 30000 )
ierr = 9;
return;
end
if ( phi < 0.01 )
ierr = 10;
return;
end
% Get fuel composition information
[ alpha, beta, gamma, delta ] = fuel( ifuel, T );
Computer Programs
% Equilibrium constant curve fit coefficients.
% Valid in range: 600 K < T < 4000 K
%
Ai
Bi
Kp = [ [
0.432168,
-0.112464e+5,
Ci
Di
Ei
0.267269e+1, -0.745744e-4,
...
0.321779e+1, -0.738336e-4,
...
0.242484e-8 ]; ...
[
0.310805,
-0.129540e+5,
0.344645e-8 ]; ...
[ -0.141784,
-0.213308e+4,
0.853461,
0.355015e-4, ...
0.646096,
0.272805e-5, ...
-0.310227e-8 ]; ...
[
0.150879e-1, -0.470959e+4,
-0.154444e-8 ]; ...
[ -0.752364,
0.124210e+5, -0.260286e+1,
0.259556e-3, ...
-0.162687e-7 ]; ...
[ -0.415302e-2,
0.148627e+5, -0.475746e+1,
0.124699e-3, ...
-0.900227e-8 ] ];
K = zeros(6,1);
for i=1:6
log10ki = Kp(i,1)*log(T/1000) + Kp(i,2)/T
+
Kp(i,3) + Kp(i,4)*T + ...
Kp(i,5)*T*T;
K(i) = 10ˆlog10ki;
end
c1 = K(1)/sqp;
c2 = K(2)/sqp;
c3 = K(3);
c4 = K(4);
c5 = K(5)*sqp;
c6 = K(6)*sqp;
[ierr,y3,y4,y5,y6 ] = guess( T, phi, alpha, beta, gamma, delta, c5, c6 );
if ( ierr ˜= 0 )
return;
end
a_s = alpha + beta/4 - gamma/2;
D1 = beta/alpha;
D2 = gamma/alpha + 2*a_s/(alpha*phi);
D3 = delta/alpha + 2*3.7619047619*a_s/(alpha*phi);
A = zeros(4,4);
final = 0;
for jj=1:MaxIter,
sqy6 = sqrt(y6);
sqy4 = sqrt(y4);
sqy3 = sqrt(y3);
y7=
c1*sqy6;
y8=
c2*sqy4;
y9=
c3*sqy4*sqy6;
y10= c4*sqy4*sqy3;
y2=
c5*sqy4*y6;
y1=
c6*sqy4*y5;
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Computer Programs
d76 = 0.5*c1/sqy6;
d84 = 0.5*c2/sqy4;
d94 = 0.5*c3*sqy6/sqy4;
d96 = 0.5*c3*sqy4/sqy6;
d103 = 0.5*c4*sqy4/sqy3;
d104 = 0.5*c4*sqy3/sqy4;
d24 = 0.5*c5*y6/sqy4;
d26 = c5*sqy4;
d14 = 0.5*c6*y5/sqy4;
d15 = c6*sqy4;
% form the Jacobian matrix
A = [ [ 1+d103,
d14+d24+1+d84+d104+d94,
d15+1,
d26+1+d76+d96 ]; ...
[ 0, 2.*d24+d94-D1*d14, -D1*d15-D1, 2*d26+2+d76+d96; ]; ...
[ d103, 2*d14+d24+2+d84+d94+d104-D2*d14,2*d15+1-D2*d15-D2, d26+d96 ]; ...
[ 2+d103, d104-D3*d14, -D3*d15-D3,0 ] ];
if ( final )
break;
end
B = [ -(y1+y2+y3+y4+y5+y6+y7+y8+y9+y10-1);
...
-(2.*y2 + 2.*y6 + y7 + y9 -D1*y1 -D1*y5); ...
-(2.*y1 + y2 +2.*y4 + y5 + y8 + y9 + y10 -D2*y1 -D2*y5); ...
-(2.*y3 + y10 -D3*y1 -D3*y5)
];
[ B, ierr ] = gauss( A, B );
if ( ierr ˜= 0 )
return;
end
y3 = y3 + B(1);
y4 = y4 + B(2);
y5 = y5 + B(3);
y6 = y6 + B(4);
nck = 0;
if ( abs(B(1)/y3) > prec )
nck = nck+1;
end
if ( abs(B(2)/y4) > prec )
nck = nck+1;
end
if ( abs(B(3)/y5) > prec )
nck = nck+1;
end
if ( abs(B(4)/y6) > prec )
nck = nck+1;
end
if( nck == 0 )
% perform top half of loop to update remaining mole fractions
% and Jacobian matrix
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final = 1;
continue;
end
end
if (jj>=MaxIter)
ierr = 3;
return;
end
Y = [ y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 ];
% consistency check
if( abs( sum(Y)-1 ) > 0.0000001 )
ierr = 4;
return;
end
% constants for partial derivatives of properties
dkdt = zeros(6,1);
for i=1:6,
dkdt(i)=2.302585*K(i)*( Kp(i,1)/T - Kp(i,2)/(T*T)+ Kp(i,4)+2*Kp(i,5)*T );
end
dcdt = zeros(6,1);
dcdt(1) = dkdt(1)/sqp;
dcdt(2) = dkdt(2)/sqp;
dcdt(3) = dkdt(3);
dcdt(4) = dkdt(4);
dcdt(5) = dkdt(5)*sqp;
dcdt(6) = dkdt(6)*sqp;
dcdp = zeros(6,1);
dcdp(1) = -0.5*c1/P;
dcdp(2) = -0.5*c2/P;
dcdp(5) = 0.5*c5/P;
dcdp(6) =
0.5*c6/P;
x1
= Y(1)/c6;
x2
= Y(2)/c5;
x7
= Y(7)/c1;
x8
= Y(8)/c2;
x9
= Y(9)/c3;
x10 = Y(10)/c4;
dfdt(1) = dcdt(6)*x1 + dcdt(5)*x2 + dcdt(1)*x7 +dcdt(2)*x8 +dcdt(3)*x9 + ...
dcdt(4)*x10;
dfdt(2) = 2.*dcdt(5)*x2 + dcdt(1)*x7 + dcdt(3)*x9 -D1*dcdt(6)*x1;
dfdt(3) = 2.*dcdt(6)*x1+dcdt(5)*x2+dcdt(2)*x8+dcdt(3)*x9+dcdt(4)*x10 - ...
D2*dcdt(6)*x1;
dfdt(4) = dcdt(4)*x10 -D3*dcdt(6)*x1;
dfdp(1) = dcdp(6)*x1 + dcdp(5)*x2 + dcdp(1)*x7 +dcdp(2)*x8;
dfdp(2) = 2.*dcdp(5)*x2 + dcdp(1)*x7
-D1*dcdp(6)*x1;
dfdp(3) = 2.*dcdp(6)*x1 + dcdp(5)*x2 + dcdp(2)*x8
dfdp(4) = -D3*dcdp(6)*x1;
- D2*dcdp(6)*x1;
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Computer Programs
dfdphi(1) = 0;
dfdphi(2) = 0;
dfdphi(3) = 2*a_s/(alpha*phi*phi)*(Y(1)+Y(5));
dfdphi(4) = 2*3.7619047619*a_s/(alpha*phi*phi)*(Y(1)+Y(5));
% solve matrix equations for independent temperature derivatives
b = -1.0 .* dfdt’; %element by element mult.
[b, ierr] = gauss(A,b);%
solve for new b with t derivatives
if ( ierr ˜= 0 )
return;
end
dydt(3) = b(1);
dydt(4) = b(2);
dydt(5) = b(3);
dydt(6) = b(4);
dydt(1) = sqrt(Y(4))*Y(5)*dcdt(6) + d14*dydt(4) + d15*dydt(5);
dydt(2) = sqrt(Y(4))*Y(6)*dcdt(5) + d24*dydt(4) + d26*dydt(6);
dydt(7) = sqrt(Y(6))*dcdt(1) + d76*dydt(6);
dydt(8) = sqrt(Y(4))*dcdt(2) + d84*dydt(4);
dydt(9) = sqrt(Y(4)*Y(6))*dcdt(3) + d94*dydt(4) + d96*dydt(6);
dydt(10) = sqrt(Y(4)*Y(3))*dcdt(4) + d104*dydt(4) + d103*dydt(3);
% solve matrix equations for independent pressure derivatives
b = -1.0 .* dfdp’; %element by element mult.
[b,ierr] = gauss(A,b); %
solve for new b with p derivatives
if ( ierr˜=0 )
return;
end
dydp(3) = b(1);
dydp(4) = b(2);
dydp(5) = b(3);
dydp(6) = b(4);
dydp(1) = sqrt(Y(4))*Y(5)*dcdp(6) + d14*dydp(4) + d15*dydp(5);
dydp(2) = sqrt(Y(4))*Y(6)*dcdp(5) + d24*dydp(4) + d26*dydp(6);
dydp(7) = sqrt(Y(6))*dcdp(1) + d76*dydp(6);
dydp(8) = sqrt(Y(4))*dcdp(2) + d84*dydp(4);
dydp(9) = d94*dydp(4) + d96*dydp(6);
dydp(10)= d104*dydp(4) + d103*dydp(3);
% molecular weights of constituents (g/mol)
%
CO2
O
H2O
OH
Mi = [ 44.01,
16.,
N2
O2
CO
H2
H
28.013,
32.00,
28.01,
2.016,
1.009,
...
NO
18.02,
17.009,
...
30.004];
if ( T > 1000 )
% high temp curve fit coefficients for thermodynamic properties ...
1000 < T < 3000 K
AAC = [ ...
[.446080e+1,.309817e-2,-.123925e-5,.227413e-9, -.155259e-13, ...
-.489614e+5,-.986359 ];
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Computer Programs
[.271676e+1,.294513e-2,-.802243e-6,.102266e-9, -.484721e-14, ...
-.299058e+5,.663056e+1 ];
[.289631e+1,.151548e-2,-.572352e-6,.998073e-10,-.652235e-14, ...
-.905861e+3,.616151e+1 ];
[.362195e+1,.736182e-3,-.196522e-6,.362015e-10,-.289456e-14,...
-.120198e+4,.361509e+1 ];
[.298406e+1,.148913e-2,-.578996e-6,.103645e-9, -.693535e-14,...
-.142452e+5,.634791e+1 ];
[.310019e+1,.511194e-3, .526442e-7,-.349099e-10,.369453e-14,...
-.877380e+3,-.196294e+1 ];
[.25e+1,0,0,0,0,.254716e+5,-.460117 ];
[.254205e+1,-.275506e-4,-.310280e-8,.455106e-11,-.436805e-15,...
.292308e+5,.492030e+1 ];
[.291064e+1,.959316e-3,-.194417e-6,.137566e-10,.142245e-15,...
.
393538e+4,.544234e+1 ];
[.3189e+1
,.133822e-2,-.528993e-6,.959193e-10,-.648479e-14,...
.982832e+4,.674581e+1 ]; ];
elseif ( T <= 1000 )
% low temp curve fit coefficients for thermodynamic properties,...
300 < T <= 1000 K
AAC = [ ...
[ 0.24007797e+1,
0.87350957e-2, -0.66070878e-5,
0.63274039e-15,
-0.48377527e+5,
[ 0.40701275e+1, -0.11084499e-2,
0.80702103e-12,
0.41521180e-5, -0.29637404e-8,
-0.30279722e+5, -0.32270046
[ 0.36748261e+1, -0.12081500e-2,
[ 0.36255985e+1, -0.18782184e-2,
0.23580424e+1 ];
0.43052778e+1 ];
0.2955535e+1
0.26765200e-2, -0.58099162e-5,
-0.46011762e+0 ];
0,
];
0,
[ 0.38375943e+1, -0.10778858e-2,
0.29639949e+1 ];
0.96830378e-6,
-0.22571094e-12, 0.36412823e+4,
[ 0.40459521e+1, -0.34181783e-2,
% initialize h, etc to zero
MW = 0;
Cp = 0;
...
% O
0.18713972e-9, ....
0.49370009e+0 ];
% OH
0.79819190e-5, -0.61139316e-8,
0.15919076e-11, 0.97453934e+4,
% Compute cp,h,s
% H2
0.25471627e+5, ...
0.24210316e-5, -0.16028432e-8,
0.38906964e-12, 0.29147644e+5,
];
0,
% CO
% H
[ 0.29464287e+1, -0.16381665e-2,
end
...
% O2
0.55210391e-8, ...
-0.18122739e-11, -0.98890474e+3, -0.22997056e+1 ];
0,
% N2
0.36923594e-5, -0.20319674e-8, ...
0.23953344e-12, -0.14356310e+5,
[ 0.25000000e+1,
...
% H2O
0.70554544e-5, -0.67635137e-8,
0.21555993e-11, -0.10475226e+4,
[ 0.37100928e+1, -0.16190964e-2,
];
0.23240102e-5, -0.63217559e-9, ...
-0.22577253e-12, -0.10611588e+4,
[ 0.30574451e+1,
0.20021861e-8, ...
0.96951457e+1 ]; % CO2
0.29974988e+1 ];
% H2
...
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Computer Programs
h = 0;
s = 0;
dMWdT = 0;
dMWdP = 0;
for i=1:10,
cpo = AAC(i,1) + AAC(i,2)*T + AAC(i,3)*Tˆ2 + AAC(i,4)*Tˆ3 + AAC(i,5)*Tˆ4;
ho = AAC(i,1) + AAC(i,2)/2*T + AAC(i,3)/3*Tˆ2 + AAC(i,4)/4*Tˆ3 + ...
AAC(i,5)/5*Tˆ4 + AAC(i,6)/T;
so = AAC(i,1)*log(T) + AAC(i,2)*T + AAC(i,3)/2*Tˆ2 + AAC(i,4)/3*Tˆ3 + ...
AAC(i,5)/4*Tˆ4 +AAC(i,7);
h = h + ho*Y(i); % h is h/rt here
MW = MW + Mi(i)*Y(i);
dMWdT = dMWdT + Mi(i)*dydt(i);
dMWdP = dMWdP + Mi(i)*dydp(i);
Cp = Cp+Y(i)*cpo + ho*T*dydt(i);
if (Y(i)> 1.0e-37)
s = s + Y(i)*(so - log(Y(i)));
end
end
R = 8.31434/MW;
v = R*T/P;
Cp = R*(Cp - h*T*dMWdT/MW);
h = h*R*T;
s = R*(-log(PATM) + s);
u=h-R*T;
dvdT = v/T*(1 - T*dMWdT/MW);
dvdP = v/P*(-1 + P*dMWdP/MW);
ierr = 0;
return;
function [ierr,y3,y4,y5,y6] = guess(T,phi,alpha,beta,gamma,delta,c5,c6)
ierr = 0;
y3 = 0;
y4 = 0;
y5 = 0;
y6 = 0;
% estimate number of total moles produced, N
n = zeros(6,1);
% Calculate stoichiometric molar air-fuel ratio
a_s = alpha + beta/4 - gamma/2;
if ( phi <= 1 )
% lean combustion
n(1) = alpha;
n(2) = beta/2;
n(3) = delta/2 + 3.76*a_s/phi;
n(4) = a_s*(1/phi - 1);
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435
else
% rich combustion
d1 = 2*a_s*(1-1/phi);
z = T/1000;
KK = exp( 2.743 - 1.761/z - 1.611/zˆ2 + 0.2803/zˆ3 );
aa = 1-KK;
bb = beta/2 + alpha*KK - d1*(1-KK);
cc = -alpha*d1*KK;
n(5) = (-bb + sqrt(bbˆ2 - 4*aa*cc))/(2*aa);
n(1) = alpha - n(5);
n(2) = beta/2 - d1 + n(5);
n(3) = delta/2 + 3.76*a_s/phi;
n(6) = d1 - n(5);
end
% total product moles per 1 mole fuel
N = sum(n);
% try to get close to a reasonable value of ox mole fraction
% by finding zero crossing of ’f’ function
ox = 1;
nIterMax=40;
for ii=1:nIterMax,
f = 2*N*ox - gamma - (2*a_s)/phi + (alpha*(2*c6*oxˆ(1/2) + 1))/ ...
(c6*oxˆ(1/2) + 1) + (beta*c5*oxˆ(1/2))/(2*c5*oxˆ(1/2) + 2);
if ( f < 0 )
break;
else
ox = ox*0.1;
if ( ox < 1e-37 )
ierr = 5;
return;
end
end
end
% now zero in on the ox mole fraction using Newton-Raphson iteration
for ii=1:nIterMax,
f = 2*N*ox - gamma - (2*a_s)/phi + (alpha*(2*c6*oxˆ(1/2) + 1))/ ...
(c6*oxˆ(1/2) + 1) + (beta*c5*oxˆ(1/2))/(2*c5*oxˆ(1/2) + 2);
df = 2*N - (beta*c5ˆ2)/(2*c5*oxˆ(1/2) + 2)ˆ2 + (alpha*c6)/ ...
(oxˆ(1/2)*(c6*oxˆ(1/2) + 1)) +(beta*c5)/ ...
(2*oxˆ(1/2)*(2*c5*oxˆ(1/2) + 2)) - (alpha*c6*(2*c6*oxˆ(1/2) + 1))/ ...
(2*oxˆ(1/2)*(c6*oxˆ(1/2) + 1)ˆ2);
dox = f/df;
ox = ox - dox;
if ( ox < 0.0 )
ierr = 6;
return;
end
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Computer Programs
if ( abs(dox/ox) < 0.001 )
break;
end
end
if( ii == nIterMax )
ierr = 7;
return;
end
y3 = 0.5*(delta + a_s/phi*2*3.76)/N;
y4 = ox;
y5 = alpha/N/(1+c6*sqrt(ox));
y6 = beta/2/N/(1+c5*sqrt(ox));
end % guess
function [B, IERQ] = gauss( A, B )
% maximum pivot gaussian elimination routine adapted
% from FORTRAN in Olikara & Borman, SAE 750468, 1975
% not using built-in MATLABⓇ routines because they issue
% lots of warnings for close to singular matrices
% that haven’t seemed to cause problems in this application
% routine below does check however for true singularity
IERQ = 0;
for N=1:3,
NP1=N+1;
BIG = abs( A(N,N) );
if ( BIG < 1.0e-05)
IBIG=N;
for I=NP1:4,
if( abs(A(I,N)) <= BIG )
continue;
end
BIG = abs(A(I,N));
IBIG = I;
end
if(BIG <= 0.)
IERQ=2;
return;
end
if( IBIG ˜= N)
for J=N:4,
TERM = A(N,J);
A(N,J) = A(IBIG,J);
A(IBIG,J) = TERM;
end
TERM = B(N);
B(N) = B(IBIG);
B(IBIG) = TERM;
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end
end
for I=NP1:4,
TERM = A(I,N)/A(N,N);
for J=NP1:4,
A(I,J) = A(I,J)-A(N,J)*TERM;
end
B(I) = B(I)-B(N)*TERM;
end
end
if( abs(A(4,4)) > 0.0 )
B(4) = B(4)/A(4,4);
B(3) = (B(3)-A(3,4)*B(4))/A(3,3);
B(2) = (B(2)-A(2,3)*B(3)-A(2,4)*B(4))/A(2,2);
B(1) = (B(1)-A(1,2)*B(2)-A(1,3)*B(3)-A(1,4)*B(4))/A(1,1);
else
IERQ=1; % singular matrix
return;
end
end % gauss()
end % ecp()
F.12 ADIABATICFLAMETEMP.M
% Computes const pressure adiabatic flame temperature T2
% from first law: dh = q
% Inputs:
T1 = 600;% initial temperature (K)
P1 = 100; % initial pressure (kPa)
PHI = 1.0; %
f = .1; %
equivalence ratio
residual fraction
ifuel=2; % 1=Methane, 2=Gasoline, 3=Diesel, 4=Methanol, 5=Nitromethane
% use FARG for initial calc of H1
[˜, H1, ˜, ˜, ˜, ˜, CP, ˜, ˜, ˜] = farg( T1, P1, PHI, f, ifuel );
fprintf(’ Initial Enthalpy H1 =
%7.1f Initial CP = %7.3f \n’,H1,CP);
P2 = P1;
T2 = 2000; %initial guess of flame temp
MAXITS = 50;
TOL = 0.00001;
for i=1:MAXITS,
[˜,˜, H2, ˜, ˜, ˜, ˜, CP, ˜, ˜, ˜] = ecp( T2, P2, PHI, ifuel );
%fprintf(’ Iterated Enthalpy H2 =
%7.1f Iterated CP = %7.3f \n’,H2,CP);
DELT2 = (H1-H2)/CP;
T2 = T2 + DELT2;
fprintf(’ Iterated Adiabatic Flame Temp (K) =
if ( abs(DELT2)/T2 < TOL )
break;
%8.1f
\n’,T2);
438
Computer Programs
end
end
fprintf(’ Final Adiabatic Flame Temp (K) =
%8.1f
\n’,T2);
F.13 OTTOFUEL.M
%program OttoFuel - computes const vol fuel air cycle
% first,isentropic compression from v1 to known v2
%establish initial conditions at state 1
clear;
T1 = 350; %Kelvin
P1 = 101.3; %kPa
phi = 0.95; %equivalence ratio
f= .05; %residual fraction,
rc=13.; %compression ratio
fuel_id=3 ; %id:1=Methane,2=Gasoline,3=Diesel,4=Methanol,5=Nitromethane
[ ˜, ˜, ˜, ˜, ˜, ˜, ˜, ˜, FS, ac ]=fuel(fuel_id, T1);
% FS is stoichiometric fuel/air ratio
% ac is available energy of combustion kJ/kg
maxits = 50;
tol =0.0001;
% call farg to get properties at 1
[y1, h1,u1, s1, v1, r, cp1, mw, dvdT, dvdP]=farg(T1,P1,phi,f,fuel_id);
%initial estimates of T2,P2
v2=v1/rc;
s2=s1;
cv1=cp1+ T1*(dvdTˆ2)/dvdP;
gam= cp1/cv1;
T2=T1*(v1/v2)ˆ(gam-1.);
P2=P1*(v1/v2)ˆgam;
%do the iteration to find T2 and P2
for i2 = 1:maxits,
[y2, h2,u2, s2, v2, r, cp2, mw, dvdT, dvdP]=farg(T2,P2,phi,f,fuel_id);
f1=s1-s2;
f2=v1/rc - v2;
det= cp2*dvdP/T2 + dvdTˆ2;
dt=(dvdP*f1 + dvdT*f2)/det;
dp= (-dvdT*f1 + cp2/T2*f2)/det;
%update T2 and P2
T2=T2 + dt;
P2=P2 + dp;
%check for convergence
if ( abs(dt)/T2 < tol && abs(dp)/P2 < tol )
break;
end
end
w12=-(u2-u1);% compression work
Computer Programs
%combustion from 2-3 with v and u constant
%initial estimates of T3,P3 at state 3
T3=3000;%Kelvin
P3=7000; % kPa
%do the iteration to find T3 and P3
for i3 = 1:maxits,
[ierr,y3, h3, u3, s3, v3, r, cp3, mw, dvdT, dvdP]=ecp(T3,P3,phi,fuel_id);
% fprintf(’ \n combustion ierr= %6.2f
\n’, ierr);
f1= u2-u3;
f2= v2-v3;
det= cp3*dvdP+T3*dvdTˆ2;
dt= (-f1*dvdP - f2*(dvdT+dvdP))/det;
dp= ((P3*dvdT-cp3)*f2 + f1*dvdT)/det;
%update T3 and P3
T3=T3 - dt;
P3=P3 - dp;
%check for convergence
if ( abs(dt)/T3 < tol && abs(dp)/P3 < tol )
break;
end
end
% isentropic expansion from v3 to known v4
%initial estimates of T4,P4
v4=v1;
cv3=cp3+ T3*(dvdTˆ2)/dvdP;
gam= cp3/cv3;
T4=T3*(v3/v4)ˆ(gam-1.);
P4=P3*(v3/v4)ˆgam;
%do the iteration to find T4 and P4
for i4 = 1:maxits,
[ierr, y4, h4, u4, s4, v4, r, cp4, mw, dvdT, dvdP]=ecp(T4,P4,phi,fuel_id);
% fprintf(’ \n expansion ierr= %6.2f
\n’, ierr);
f1=s3-s4;
f2=rc*v3 - v4;
det= cp4*dvdP/T4 + dvdTˆ2;
dt=(dvdP*f1 + dvdT*f2)/det;
dp= (-dvdT*f1 + cp4/T4*f2)/det;
%update T4 and P4
T4=T4 + dt;
P4=P4 + dp;
%check for convergence
if ( abs(dt)/T4 < tol && abs(dp)/P4 < tol )
break;
end
end
%compute cycle parameters
w=u1-u4;% net work
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440
Computer Programs
imep=w/(v1-v2); %imep
eta=w*(1+phi*FS)/phi/FS/(1.-f)/ac;
%output state and cycle parameters
fprintf(’ \n Ottofuel input conditions:phi= %6.2f fuel= %4d \n’,phi,fuel_id);
fprintf(’State \t\t
fprintf(’Pressure (kPa)=
1 \t
2 \t \t
3\t
\t
4 \n’)
%7.1f \t %7.1f \t %7.1f \t %7.1f \n’,P1,P2,P3,P4);
fprintf(’Temperature (K)= %7.1f \t %7.1f \t %7.1f \t %7.1f
\n’,T1,T2,T3,T4);
fprintf(’Enthalpy(kJ/kg)= %7.1f \t %7.1f \t %7.1f \t %7.1f
\n’,h1,h2,h3,h4);
fprintf(’Int.Energy(kJ/kg)=%7.1f \t %7.1f \t %7.1f \t %7.1f
\n’,u1,u2,u3,u4);
fprintf(’Volume (mˆ3/kg) =%7.3f \t %7.3f \t %7.3f \t %7.3f
\n’, v1,v2,v3,v4);
fprintf(’Cp (kJ/kg K) =
\n \n’, ...
%7.3f \t %7.3f \t %7.3f \t %7.3f
cp1,cp2,cp3,cp4);
fprintf(’ Work (kJ/kg)= %7.1f \n’, w);
fprintf(’ Efficiency= %7.3f \n’, eta);
fprintf(’ Imep (kPa)= %7.1f \n \n’, imep);
fprintf(’ Iterations = \t \t %4d \t %4d \t %4d
\n’, i2,i3,i4);
F.14 FOURSTROKEFUELAIR.M
%program four stroke ottofuel - computes const vol fuel air cycle
%establish initial conditions for intake stroke
clear;
Ti = 300; % intake temperature (K)
Pi = 52.5; % intake pressure (kPa)
Pe = 105; % exhaust pressure (kPa)
phi = .8; % equivalence ratio
rc = 10.; % compression ratio
%get specific heat input qin and stoichiometric fuel-air ratio FS
fuel_id=2 ; %id: 1=Methane, 2=Gasoline, 3=Diesel, 4=Methanol, 5=Nitromethane
[ ˜, ˜, ˜, ˜, ˜, ˜, ˜, ˜, FS, qin ] = fuel( fuel_id, Ti );
% find enthalpy hi of intake fuel-air mixture
ff=0; % no residual fraction in intake air
[yi,hi,ui, si, vi, r, cpi, mw, dvdT, dvdP] = farg( Ti, Pi, phi, ff, fuel_id );
maxits = 100;
tol =0.0001;
%initial estimates of residual fraction and initial temp
f= .1; % residual fraction,
T1=350.; % initial temperature
P1=Pi; % no pressure drop in intake
% main iteration loop around cycle to get converged values of f and T1
for imain = 1:maxits,
% isentropic compression from v1 to known v2
% call farg to get properties at 1
[y1,h1,u1, s1, v1, r, cp1, mw, dvdT, dvdP] = farg( T1, P1, phi, f, fuel_id );
%initial estimates of T2,P2
v2=v1/rc;
s2=s1;
Computer Programs
cv1=cp1+ T1*(dvdTˆ2)/dvdP;
gam= cp1/cv1;
T2=T1*(v1/v2)ˆ(gam-1.);
P2=P1*(v1/v2)ˆgam;
%do the iteration to get T2 and P2 at end of compression
for i2 = 1:maxits,
[y2, h2,u2, s2, v2, r, cp2, mw, dvdT, dvdP]=farg(T2,P2,phi,f,fuel_id);
f1=s1-s2;
f2=v1/rc - v2;
det= cp2/T2*dvdP + dvdTˆ2;
dt=(dvdP*f1 + dvdT*f2)/det;
dp= (-dvdT*f1 + cp2/T2*f2)/det;
%update T2 and P2
T2=T2 + dt;
P2=P2 + dp;
%check for convergence
%check for convergence
if ( abs(dt)/T2 < tol && abs(dp)/P2 < tol )
break;
end
end
w12=-(u2-u1);% compression work
% combustion from 2-3 with v and u constant
%initial estimates of T3,P3 at state 3
T3=3000;%Kelvin
P3=7000; % kPa
%do the iteration to get T3 and P3
for i3 = 1:maxits,
[ierr,y3,h3,u3,s3,v3,r,cp3,mw,dvdT,dvdP] =ecp(T3,P3,phi,fuel_id);
f1= u2-u3;
f2= v2-v3;
det= cp3*dvdP + T3*dvdTˆ2;
dt= (-f1*dvdP - f2*(dvdT+dvdP))/det;
dp= ((P3*dvdT-cp3)*f2 + f1*dvdT)/det;
%update T3 and P3
T3=T3 - dt;
P3=P3 - dp;
%check for convergence
%check for convergence
if ( abs(dt)/T3 < tol && abs(dp)/P3 < tol )
break;
end
end
% isentropic expansion of combustion products from v3 to known v4
% initial estimates of T4,P4
v4=v1;
cv3=cp3+ P3*v3/T3*(dvdTˆ2)/dvdP;
441
442
Computer Programs
gam= cp3/cv3;
T4=T3*(v3/v4)ˆ(gam-1.);
P4=P3*(v3/v4)ˆgam;
%do the iteration to get T4 and P4
for i4 = 1:maxits,
[ierr,y4,h4,u4,s4,v4,r,cp4,mw,dvdT,dvdP]=ecp(T4,P4,phi,fuel_id);
f1=s3-s4;
f2=rc*v3 - v4;
det= cp4*dvdP/T4 + dvdTˆ2;
dt=(dvdP*f1 + dvdT*f2)/det;
dp= (-dvdT*f1 + cp4/T4*f2)/det;
%update T4 and P4
T4=T4 + dt;
P4=P4 + dp;
%check for convergence
if ( abs(dt)/T4 < tol && abs(dp)/P4 < tol )
break;
end
end
% isentropic blowdown of control mass to exhaust pressure
P5=Pe;
s5=s4;
%initial estimates of T5
cv4=cp4+ P4*v4/T4*(dvdTˆ2)/dvdP;
gam= cp4/cv4;
T5=T4*(P5/P4)ˆ((gam-1.)/gam);
% do iteration for T5
for i5 = 1:maxits,
[ierr,y5,h5,u5,s5,v5,r,cp5,mw,dvdT,dvdP]=ecp(T5,P5,phi,fuel_id);
f1=s4-s5;
dt=T5*f1/cp5;
T5=T5 + dt;
%check for convergence
if ( abs(dt)/T5 < tol
)
break;
end
end
% recompute residual fraction
fold = f;
v6=v5;
f=v4/v6/rc;
% constant pressure exhaust stroke
h6=h5;
%recompute h1 with fuel-air mixture and new residual fraction
h1= f*(h6 + (Pi-Pe)*v6) + (1-f)*hi;
T1old=T1;
%recompute T1 with latest f and h1
Computer Programs
443
for i6 = 1:maxits,
[y1,h1new,u1,s1,v1,r,cp1,mw,dvdT,dvdP]=farg(T1 Pi,phi,f,fuel_id);
g=h1new-h1;
dt=-g/cp1;
T1=T1+dt;
%check for convergence
if ( abs(dt)/T1 < tol )
break;
end
end
%check for convergence of main iteration loop
dt=T1old - T1;
df=fold - f;
if ( abs(dt)/T1 < tol && abs(df)/f < tol )
break;
end
end %end of main iteration loop
%compute cycle parameters
w = u1-u4;% net work
imep = w/(v1-v2); %imep
eta = w*(1+phi*FS)/phi/FS/(1.-f)/qin; % thermal efficiency
pmep = Pe -Pi; %pmep
etanet = eta*(1.- pmep/imep); %net thermal efficiency
ev = rc*(1.-f)*vi/(rc-1.)/v1;
%output state and cycle parameters
fprintf(’ \n Ottofuel inlet: Temp (K)= %5.1f
Pressure (kPa)= %5.1f
...
phi= %6.2f fuel= %3d \n’, Ti, Pi, phi, fuel_id );
fprintf(’ State \t\t
1 \t
2 \t \t
3\t
\t
4 \n’)
fprintf(’Pressure (kPa)= %7.1f \t %7.1f \t %7.1f \t %7.1f \n’,...
P1,P2,P3,P4);
fprintf(’Temperature (K)=%7.1f \t %7.1f \t %7.1f \t %7.1f
\n’,...
T1,T2,T3,T4);
fprintf(’Enthalpy(kJ/kgK)=%7.1f \t %7.1f \t %7.1f \t %7.1f
\n’,...
h1,h2,h3,h4);
fprintf(’Int. Energy(kJ/kg)=%6.1f \t %7.1f \t %7.1f \t %7.1f
\n’,...
u1,u2,u3,u4);
fprintf(’Volume (mˆ3/kg)= %7.3f \t %7.3f \t %7.3f \t %7.3f
\n’, ....
v1,v2,v3,v4);
fprintf(’Entropy(kJ/kgK)= %6.3f \t %7.3f \t %7.3f \t %7.3f
\n’, ...
s1,s2,s3,s4);
fprintf(’Cp (kJ/kg K) = %7.3f \t %7.3f \t %7.3f \t %7.3f
\n \n’, ...
cp1,cp2,cp3,cp4);
fprintf(’Work (kJ/kg)= %7.1f
\t \t Volumetric Efficiency=
fprintf(’Ideal Thermal Efficiency= %7.3f
%7.4f \n’, w, ev);
\t Net Thermal Efficiency= %7.4f ...
\n’, eta, etanet);
fprintf(’Imep (kPa)= %7.1f \t \t
\t Pmep (kPa)= %7.1f \n’, imep, pmep);
fprintf(’Exhaust Temperature (K)= %7.1f \t Residual Mass Fraction ...
444
Computer Programs
f =%7.4f \n’, T5,f);
fprintf(’Iterations = \t %4d\t %4d \t %4d \t %4d
\n’, imain,i2,i3,i4);
F.15 HOMOGENEOUS.M
function [ ETA, IMEP, NOX_ppm ] = Homogeneous(varargin)
% Two Zone Arbitrary Heat Release (Fuel Inducted Engine)
%
%
Find:
% 1. Indicated thermal efficiency - ETA
% 2. Indicated mean effective pressure - IMEP (kPa)
%
%
Note:
% 1. Cosine burning law is employed
% 2. Fuel is gasoline, C7H17
R = 7; % Compression ratio - R
B = .0925; % Bore - B (m)
S = .1143; % Stroke - S (m)
EPS = 0.25; % Half stroke to rod ratio - EPS
RPM = 1200; % Engine speed - RPM
HEAT = 500; % Heat transfer coefficient
BLOWBY = 0.8; % Blowby coefficient
THETAS = -15; % Start of heat release (deg ATDC)
THETAB = 45; % Burn angle (deg)
PHI = 0.88; % Equivalence ratio - PHI
F = 0.05; % Residual fraction - F
TW = 400; % Wall temperature - TW
fuel_type = 2; % gasoline
FS = 0.06548; % stoichiometric fuel-air ratio for gasoline
A0 = 47870; %
T1 = 363; %K
P1 = 100; % kPa
if ( nargin == 3 )
PHI = varargin{1};
F = varargin{2};
RPM = varargin{3};
end
OMEGA = RPM*pi/30;
to_ppm = 10ˆ6; % convert from mass fraction to ppm
MW_NO = 30;
% molecular weight of NO, g/mol
THETA = -180;
DTHETA = 1;
THETAE = THETA+DTHETA;
[ VOL, X, EM ] = auxiliary( THETA );
NNOX = THETAB/DTHETA;
NY = 6+NNOX;
Y = zeros(NY,1);
Computer Programs
445
Y(1) = P1;
Y(2) = nan;
Y(3) = T1;
[˜, ˜, ˜, ˜, vU, ˜, ˜, ˜, ˜, ˜] = farg( Y(3), Y(1), PHI, F, fuel_type );
MNOT = VOL/vU;
M = EM*MNOT;
NN = 36*10;
SAVE.THETA = zeros( NN, 1 );
SAVE.VOL = zeros( NN, 1 );
SAVE.T = zeros(NN, 1 );
SAVE.P = zeros( NN, 1 );
SAVE.MDOTFI = zeros( NN, 1 );
SAVE.NOx = zeros(NN,5);
fprintf( ’THETA
HEAT LOSS
VOL
fprintf( ’ deg
H-LEAK
g
fprintf(’%7.1f
PRESS
-J
%6.1f
%3.3f
BURN TEMP
UNBURNED T
WORK
...
NOx\n’ );
cmˆ3
J
%5.2f
BURN FRAC
MASS
kPa
K
K
J
....
ppm\n’ );
%6.1f
%6.1f
%6.1f
%5.0f
%5.0f %5.3f
....
%6.1f\n’, THETA, VOL*1000000, X, Y(1), Y(2), Y(3), Y(4)*1000, ...
Y(5)*1000, M*1000, Y(6)*1000, 0.0 );
II = 1;
for III=1:36,
for JJJ=1:10,
[ Y ] = integrate( THETA, THETAE, Y );
[ VOL, X, EM ] = auxiliary( THETA );
M = EM*MNOT;
THETA=THETAE;
THETAE=THETA+DTHETA;
% save data for plotting later
SAVE.THETA(II) = THETA;
SAVE.VOL(II) = VOL;
SAVE.P(II) = Y(1);
SAVE.TB(II) = Y(2);
SAVE.TU(II) = Y(3);
SAVE.X(II) = X;
SAVE.NOX(II,:) = [ Y(6+1), Y(round(6+0.25*NNOX)),...
Y(round(6+0.5*NNOX)), Y(round(6+0.75*NNOX)), Y(6+NNOX) ]*to_ppm;
II=II+1;
if ( THETAS >= THETA && THETAS < THETAE )
Y(2) = tinitial( Y(1), Y(3), PHI, F );
end
if ( THETA > THETAS + THETAB )
Y(3) = nan;
end
end
fprintf(’%7.1f
%6.1f
%3.3f
%6.1f
%6.1f
%6.1f
%5.0f
...
446
Computer Programs
%5.0f
%5.3f
%5.2f
%6.1f\n’, ...
THETA, VOL*1000000, X, Y(1), Y(2), Y(3), Y(4)*1000, Y(5)*1000, ....
M*1000, Y(6)*1000, Y(7)*to_ppm );
end
% integrate total NOx value
NOX_ppm = 0;
for nn=1:NNOX;
THETA = THETAS + (nn-1)/(NNOX-1)*THETAB;
dxbdtheta = 0.5*sin(pi*(THETA-THETAS)/THETAB)*pi/THETAB;
dxb = dxbdtheta*DTHETA;
NOX_ppm = NOX_ppm + Y(6+nn)*dxb*to_ppm;
end
ETA = Y(4)/MNOT*(1+PHI*FS*(1-F))/PHI/FS/(1-F)/A0;
IMEP = Y(4)/(pi/4*Bˆ2*S);
fprintf(’ETA=%1.4f IMEP=%7.3f kPa NOx = %6.1f ppm\n’,ETA,IMEP,NOX_ppm );
if ( nargin == 0 )
% if not called externally with custom PHI, F, and RPM parameters,
% generate some plots
sTitle = sprintf(’Homogenous 2 zone, gasoline, PHI=%.2f F=%.2f ...
RPM=%.1f\nETA=%.3f IMEP=%.2f kPa
NOx=%.1f ppm ’, PHI, F, ...
RPM, ETA, IMEP, NOX_ppm );
figure;
plot( SAVE.THETA, SAVE.X, ’linewidth’,2 );
set(gca,’fontsize’,18,’linewidth’,2,’Xlim’,[-100 100]);
xlabel( ’\theta’,’fontsize’,18);
ylabel(’burn fraction’,’fontsize’,18);
figure;
plot(
SAVE.THETA, SAVE.P,’linewidth’,2 );
set(gca,’fontsize’,18,’linewidth’,2,’Xlim’,[-100 100]);
xlabel( ’\theta’,’fontsize’,18);
ylabel(’pressure (kPa)’,’fontsize’,18);
figure;
plot( SAVE.THETA, SAVE.TU, ’-’,SAVE.THETA, SAVE.TB,’--’,’linewidth’,2 );
set(gca,’fontsize’,18,’linewidth’,2,’Xlim’,[-100 100]);
xlabel( ’\theta’,’fontsize’,18);
ylabel( ’temperature (K)’, ’fontsize’,18);
legend(’Unburned’,’Burned’, ’Location’, ’SouthEast’);
figure;
plot( SAVE.THETA, SAVE.NOX,’linewidth’,2 );
set(gca,’fontsize’,18,’linewidth’,2,’Xlim’,[-100 100]);
xlabel(’\theta’,’fontsize’,18);
ylabel(’NOx (ppm)’,’fontsize’,18);
axis( [ THETAS, 110, 0, max(max(SAVE.NOX)*1.1) ] );
%legend( ’X=0’, ’X=0.25’, ’X=0.5’, ’X=0.75’, ’X=1’, ’Location’, ...
’SouthEast’ );
%title( sTitle );
end
Computer Programs
447
function [ TB ] = tinitial( P, TU, PHI, F )
TB = 2000;
[˜, HU,˜, ˜, ˜, ˜, ˜, ˜, ˜, ˜] = farg( TU, P, PHI, F, fuel_type );
for ITER=1:50,
[ierr, ˜, HB,˜, ˜, ˜, ˜, CP, ˜, ˜, ˜] = ecp( TB, P, PHI, fuel_type );
if ( ierr ˜= 0 )
fprintf(’Error in ECP(%g, %g, %g): %d\n’, TB, P, PHI, ierr )
end
DELT = +(HU-HB)/CP;
TB = TB + DELT;
if ( abs(DELT/TB) < 0.001 )
break;
end
end
end
function [ VOL, X, EM ] = auxiliary( THETA )
VTDC = pi/4*Bˆ2*S/(R-1); % m3
VOL = VTDC*(1 + (R-1)/2*(1-cosd(THETA) + 1/EPS*(1-sqrt(1- ...
(EPS*sind(THETA))ˆ2))));
X = 0.5*(1-cos(pi*(THETA-THETAS)/THETAB));
if ( THETA <= THETAS )
X = 0.;
end
if ( THETA >= THETAS+THETAB )
X = 1.;
end
EM = exp(-BLOWBY*(THETA*pi/180 + pi)/OMEGA);
end
function [Y] = integrate( THETA, THETAE, Y )
[TT, YY ] = ode23( @rates, [ THETA, THETAE ], Y );
for J=1:NY,
Y(J) = YY(length(TT),J);
end
function [ YPRIME ] = rates( THETA, Y )
YPRIME = zeros(NY,1);
[ VOL, X, EM ] = auxiliary( THETA );
M = EM*MNOT;
DUMB = sqrt(1-(EPS*sind(THETA))ˆ2);
DV = pi/8*Bˆ2*S*sind(THETA)*(1+EPS*cosd(THETA)/DUMB);
AA = (DV + VOL*BLOWBY/OMEGA)/M;
C1 = HEAT*(pi*Bˆ2/2 + 4*VOL/B)/OMEGA/M/1000;
C0 = sqrt(X);
P = Y(1);
TB = Y(2);
TU = Y(3);
% three different computations are required depending upon the size
% of the mass fraction burned
448
Computer Programs
if ( X > 0.999 )
%
EXPANSION
[ierr,YB,HL, ˜, ˜,VB, ˜,CP, ˜,DVDT,DVDP]=ecp(TB,P,PHI,fuel_type);
if ( ierr ˜= 0 )
fprintf(’Error in ECP(%g, %g, %g): %d\n’, TB, P, PHI, ierr );
end
BB = C1/CP*DVDT*TB*(1-TW/TB);
CC = 0;
DD = 1/CP*TB*DVDTˆ2 + DVDP;
EE = 0;
YPRIME(1) = (AA + BB + CC)/(DD + EE);
YPRIME(2) = -C1/CP*(TB-TW) + 1/CP*DVDT*TB*YPRIME(1);
YPRIME(3) = 0;
elseif ( X > 0.001 )
%
COMBUSTION
[˜,HU, ˜, ˜,VU, ˜,CPU, ˜,DVDTU,DVDPU]=farg(TU,P,PHI,F,fuel_type);
[ierr,YB,HB, ˜, ˜,VB, ˜,CPB, ˜,DVDTB,DVDPB]=ecp(TB,P,PHI,fuel_type);
if ( ierr ˜= 0 )
fprintf(’Error in ECP(%g, %g, %g): %d\n’, TB, P, PHI, ierr );
end
BB = C1*(1/CPB*TB*DVDTB*C0*(1-TW/TB) + 1/CPU*TU*DVDTU*(1-C0)* ...
(1-TW/TU));
DX = 0.5*sin( pi*(THETA-THETAS)/THETAB )*180/THETAB;
CC = -(VB-VU)*DX - DVDTB*(HU-HB)/CPB*(DX-(X-Xˆ2)*BLOWBY/OMEGA);
DD = X*(VBˆ2/CPB/TB*(TB/VB*DVDTB)ˆ2 + DVDPB);
EE = (1-X)*(1/CPU*TU*DVDTUˆ2 + DVDPU);
HL = (1-Xˆ2)*HU + Xˆ2*HB;
YPRIME(1) = (AA + BB + CC)/(DD + EE);
YPRIME(2) = -C1/CPB/C0*(TB-TW) + 1/CPB*TB*DVDTB*YPRIME(1) + ...
(HU-HB)/CPB*(DX/X - (1-X)*BLOWBY/OMEGA);
YPRIME(3) = -C1/CPU/(1+C0)*(TU-TW) + 1/CPU*TU*DVDTU*YPRIME(1);
else
%
COMPRESSION
[˜, HL, ˜, ˜, ˜, ˜,CP, ˜,DVDT,DVDP]=farg(TU,P,PHI,F,fuel_type);
BB = C1*1/CP*TU*DVDT*(1-TW/Y(3));
CC = 0;
DD = 0;
EE = 1/CP*TU*DVDTˆ2 + DVDP;
YPRIME(1) = ( AA + BB + CC )/(DD + EE);
YPRIME(2) = 0;
YPRIME(3) = -C1/CP*(Y(3)-TW) + 1/CP*Y(3)*DVDT*YPRIME(1);
end
% common to all cases
YPRIME(4) = Y(1)*DV;
YPRIME(5) = 0;
if ( ˜isnan(TB) )
YPRIME(5) = YPRIME(5) + C1*M*C0*(TB-TW);
Computer Programs
449
end
if ( ˜isnan(TU) )
YPRIME(5) = YPRIME(5) + C1*M*(1-C0)*(TU-TW);
end
YPRIME(6) = BLOWBY*M/OMEGA*HL;
% perform NOx integration for each element burned
if ( X > 0.001 )
% COMBUSTION OR EXPANSION
for k=1:NNOX,
if ( THETA >= THETAS + (k-1)/(NNOX-1)*THETAB )
% convert Y(6+k) to [NO] mol/cmˆ3 from mass fraction
% and then back
YPRIME(6+k) = zeldovich( TB, P/100, YB, Y(6+k)/(MW_NO*VB*1000) ) ...
*MW_NO*VB*1000/OMEGA;
end
end
end
% 1/omega is s/rad, so convert to s/deg
for JJ=1:NY,
YPRIME(JJ) = YPRIME(JJ)*pi/180;
end
end
end
function [ dNOdt ] = zeldovich( T, P, y, NO )
% calculate rate of NO formation d[NO]/dt given
% inputs:
%
T
[K]
: gas mixture temperature, kelvin
%
P
[bar]
: cylinder pressure, bar
%
y
[...]
: equilibrium mole fraction of constituents
%
NO [mol/cmˆ3]
: current NOx concentration
% outputs:
%
dNOdt [ (mol/cmˆ3) / sec ]
: rate of NO formation
% extended zeldovich rate constants from Heywood Table 11.1 (cmˆ3/mol-s)
k1 = 7.6*10ˆ13*exp(-38000/T);
k2r = 1.5*10ˆ9*T*exp(-19500/T);
k3r = 2*10ˆ14*exp(-23650/T);
% calculate molar concentration [mol/cmˆ3]
N_V = (100000*P)/(8.314*T)*(1/100)ˆ3;
N2e = y(3)*N_V;
He = y(7)*N_V;
Oe = y(8)*N_V;
NOe = y(10)*N_V;
R1 = k1*Oe*N2e;
R2 = k2r*NOe*Oe;
R3 = k3r*NOe*He;
alpha = NO/NOe;
dNOdt = 2*R1*(1-alpha*alpha)/(1+alpha*R1/(R2+R3));
450
Computer Programs
end
end
F.16 FRICTION.M
% program to compute friction mean effective pressure
% fmep units in kPa
% inputs
clear;
N = 3000; %engine speed rpm
b = 86; % bore (mm)
s = 86; % stroke (mm)
nc =4; % # cylinders
pin=101; % intake manifold pressure (kPa)
db = 56; % main bearing diameter (mm)
lb = 21; % main bearing length (mm)
niv = 2; % # intake valves/cyl
nev = 2; % # exhaust valves/cyl
div = 35; % intake valve diameter (mm);
dev= 31; % exhaust valve diameter (mm);
lv = 11; % valve lift (mm)
mu = 100.e-3 ; % dynamic viscosity (Pa s)
pa= 101; %atmospheric pressure (kPa)
Up = 2.* N * s/60; % mean piston speed (mm/s)
denom = nc*bˆ2*s;
nb= nc+1; %# main crankshaft bearings
nv = (niv+nev)*nc; % # valves (total)
% friction coefficients
c_cb=0.0202; % crankshaft bearing
c_cs=93600; % crankshaft seals
c_pb=0.0202; % piston bearings
c_ps=14; % piston seals
c_pr=2707; % pison ringpack
c_vb=6720; % camshaft bearings
c_vh=0.5; % oscillating hydrodynamic
c_vm=10.7; % oscillating mixed
c_vs=1.2; % seals
c_vf= 207; % flat cam follower
c_vr=0.0151;% roller cam follower
c_1o=1.28; c_2o=0.0079; c_3o=-8.4e-7; %oil pump
c_1w=0.13; c_2w=0.002; c_3w=3.e-7; %water pump
c_1f=1.72; c_2f=0.00069; c_3f=1.2e-7; %fuel injection
c_iv=4.12e-3; % inlet valves (kPa sˆ2/mˆ2)
c_ev=c_iv; %exhaust valves (kPa sˆ2/mˆ2)
c_es=0.178; % exhaust system (kPa sˆ2/mˆ2)
% component fmeps
%crankshaft
Computer Programs
f_cb=c_cb*nb*N.ˆ(0.6)*dbˆ3*lb/denom;
f_cs=c_cs*db/denom;
f_crank=f_cb+f_cs;
%piston assembly
f_pb=c_pb*nb*N.ˆ(0.6)*dbˆ3*lb/denom; % bearings
f_ps=c_ps*Up.ˆ(0.5)/b; % skirt
f_pr=c_pr*Up.ˆ(0.5)/(bˆ2); %ringpack
f_piston=f_pb+f_ps+f_pr;
%valvetrain
f_cam=c_vb*nb*N.ˆ(0.6)/denom; %bearings
f_vh=c_vh*nv*lvˆ(1.5)*N.ˆ(0.5)/denom; %oscill hydro
f_vm=c_vm*(2+10./(5+mu.*N))*lv*nv/(nc*s);
f_vs=c_vs; %seals
%cam followers - choose flat or roller
f_ff=c_vf*(2+10./(5+mu.*N))*nv/(nc*s); % flat
%f_rf=c_nv*N/(nc*s); % or roller
f_valve=f_cam+f_vh+f_vm+f_vs+f_ff;
%auxiliary
f_oil= c_1o + c_2o*N + c_3o*N.ˆ2; %oil pump
f_wat= c_1w + c_2w*N + c_3w*N.ˆ2; %water pump
f_fuel= c_1f + c_2f*N + c_3f*N.ˆ2; %fuel pump
f_aux=f_oil+f_wat+f_fuel;
%pumping
dpis= pa-pin;
dpiv=c_iv*(pin/pa.*Up/1000*bˆ2/niv/divˆ2).ˆ2;
dpev=c_ev*(pin/pa.*Up/1000*bˆ2/nev/devˆ2).ˆ2;
dpes=c_es*(pin/pa.*Up/1000).ˆ2;
f_pump=dpis+dpiv+dpev+dpes;
%total
f_tot=f_crank+f_piston+f_valve+f_aux+f_pump;
fprintf(’ \n fmep crankshaft (kPa)= %7.1f
fprintf(’ fmep piston (kPa)=
%7.1f
\n’,f_crank);
\n’,f_piston);
fprintf(’ fmep valvetrain (kPa)= %7.1f
\n’,f_valve);
fprintf(’ fmep auxiliary (kPa)=
\n’,f_aux);
%7.1f
fprintf(’ fmep pumping (kPa)=
%7.1f
\n’,f_pump);
fprintf(’ fmep total (kPa)=
%7.1f
\n’,f_tot);
F.17 WOSCHNIHEATTRANSFER.M
function [ ] = WoschniHeatTransfer( )
% Gas cycle heat release code with Woschni heat transfer
clear( );
thetas = -20; % start of heat release (deg)
thetad = 40; % duration of heat release (deg)
r =10;
% compression ratio
gamma = 1.3; % gas const
Q = 20;
% dimensionless total heat release
451
452
Computer Programs
beta = 1.5; % dimensionless volume
a = 5;
% weibe parameter a
n = 3;
% weibe exponent n
omega =200.; % engine speed rad/s
c = 0;
% mass loss coeff
s = 0.1;
% stroke (m)
b = 0.1;
% bore (m)
T_bdc = 300;
tw = 1.2;
% temp at bdc (K)
% dimensionless cylinder wall temp
P_bdc = 100;
% pressure at bdc (kPa)
Up = s*omega/pi; % mean piston speed (m/s)
step=1;
% crankangle interval for calculation/plot
NN=360/step; % number of data points
theta = -180; % initial crankangle
thetae = theta + step; % final crankangle in step
% initialize results data structure
save.theta=zeros(NN,1);
save.vol=zeros(NN,1);
% volume
save.press=zeros(NN,1); % pressure
save.work=zeros(NN,1);
% work
save.heatloss=zeros(NN,1); % heat loss
save.mass=zeros(NN,1);
% mass left
save.htcoeff=zeros(NN,1); % heat transfer coeff
save.heatflux=zeros(NN,1); % heat flux (W/mˆ2)
fy=zeros(4,1); % vector for calulated pressure, work, heat and mass loss
fy(1) = 1; % initial pressure (P/P_bdc)
fy(4) = 1; % initial mass (-)
%for loop for pressure and work calculation
for i=1:NN,
[fy, vol, ht,hflux] = integrate_ht(theta,thetae,fy);
% print values
% fprintf(’%7.1f
%7.2f
%7.2f
%7.2f \n’, theta,vol,fy(1),fy(2),fy(3));
% reset to next interval
theta = thetae;
thetae = theta+step;
save.theta(i)=theta; % put results in output vectors
save.vol(i)=vol;
save.press(i)=fy(1);
save.work(i)=fy(2);
save.heatloss(i)=fy(3);
save.mass(i)=fy(4);
save.htcoeff(i)=ht;
save.hflux(i)=hflux;
end % end of pressure and work for loop
[pmax, id_max] = max(save.press(:,1)); % find max pressure
thmax=save.theta(id_max);
% and crank angle
Computer Programs
ptdc=save.press(NN/2)/pmax;
w=save.work(NN,1);
% w is cumulative work
massloss =1- save.mass(NN,1);
eta=w/Q;
% thermal efficiency
imep = eta*Q*(r/(r -1)); %imep/P1V1
eta_rat = eta/(1-rˆ(1-gamma));
% output overall results
fprintf(’ Weibe Heat Release with Heat and Mass Loss
fprintf(’ Theta_start =
%5.2f
fprintf(’ Theta_dur =
%5.2f
\n’, thetad);
fprintf(’ P_max/P1 =
%5.2f
\n’, pmax);
fprintf(’ Theta @P_max =
%7.1f
\n’);
\n’, thetas);
\n’,thmax);
fprintf(’ Net Work/P1V1 =
%7.2f
\n’, w);
fprintf(’ Heat Loss/P1V1 =
%7.2f
\n’, save.heatloss(NN,1));
fprintf(’ Mass Loss/m =
%7.3f
fprintf(’ Efficiency =
%5.2f
\n’,massloss );
\n’, eta);
fprintf(’ Eff./Eff. Otto =
%5.2f
\n’, eta_rat);
fprintf(’ Imep/P1 =
%5.2f
\n’, imep);
%plot results
figure();
plot(save.theta,save.work,’-’,save.theta,save.heatloss,’--’,’linewidth’,2 )
set(gca, ’Xlim’,[-180 180],’fontsize’, 18,’linewidth’,1.5);
hleg1=legend(’Work’, ’Heat Loss’,’Location’,’NorthWest’);
set(hleg1,’Box’, ’off’)
xlabel(’Crank Angle \theta (deg)’,’fontsize’, 18)
ylabel(’Cumulative Work and Heat Loss’,’fontsize’, 18)
plot(save.theta,save.press,’-’,’linewidth’,2 )
set(gca, ’fontsize’, 18,’linewidth’,1.5,’Xlim’, [-180 180]);
xlabel(’Crank Angle (deg)’,’fontsize’, 18)
ylabel(’Pressure (bar)’,’fontsize’, 18)
figure();
plot(save.theta,save.htcoeff,’-’,’linewidth’,2 )
set(gca, ’fontsize’, 18,’linewidth’,1.5,’Xlim’, [-180 180]);
xlabel(’Crank Angle \theta (deg)’,’fontsize’, 18)
ylabel(’Heat transfer coefficient h (W/mˆ2-K)’,’fontsize’, 18)
figure();
plot(save.theta,save.hflux,’-’,’linewidth’,2 )
set(gca, ’fontsize’, 18,’linewidth’,1.5,’Xlim’, [-180 180]);
xlabel(’Crank Angle \theta (deg)’,’fontsize’, 18)
ylabel(’Heat flux q{"} (MW/mˆ2)’,’fontsize’, 18)
function[fy,vol,ht, hflux] = integrate_ht(theta,thetae,fy)
%
ode23 integration of the pressure differential equation
%
from theta to thetae with current values of fy as initial conditions
[tt, yy] = ode23(@rates, [theta thetae], fy);
% put last element of yy into fy vector
for j=1:4
453
454
Computer Programs
fy(j) = yy(length(tt),j);
end
% pressure differential equation
function [yprime] = rates(theta,fy)
vol=(1.+ (r -1)/2.*(1-cosd(theta)))/r;
dvol=(r - 1)/2.*sind(theta)/r*pi/180.; %dvol/dtheta
dx=0.;
if(theta>thetas) % heat release >0
dum1=(theta -thetas)/thetad;
x=1-exp(-(a*dum1ˆn));
dx=(1-x)*a*n*dum1ˆ(n-1)/thetad; %dx/dthetha
end
P=P_bdc*fy(1); %P(theta) (kPa)
T=T_bdc*fy(1)*vol; % T(theta) (K)
term4=T_bdc*(r-1)*(fy(1)-volˆ(-gamma))/r; % comb. vel. increase
U=2.28*Up + 0.00324*term4; % Woschni vel (m/s)
ht = 3.26 *Pˆ(0.8)*Uˆ(0.8)*bˆ(-0.2)*Tˆ(-0.55); %Woschni ht coeff
hflux=ht*T_bdc*(fy(1)*vol/fy(4) - tw)/10ˆ6; %heat flux MW/mˆ2
h = ht*T_bdc*4/(1000*P_bdc*omega*beta*b); %dimensionless ht coeff
term1= -gamma*fy(1)*dvol/vol;
term3= h*(1. + beta*vol)*(fy(1)*vol/fy(4) - tw)*pi/180.;
term2= (gamma-1)/vol*(Q*dx - term3);
yprime(1,1)= term1 + term2 - gamma*c/omega*fy(1)*pi/180;
yprime(2,1)= fy(1)*dvol;
yprime(3,1)= term3;
yprime(4,1)= -c*fy(4)/omega*pi/180;
end %end of function rates
end % end of function integrate_ht
end % end of function HeatReleaseHeatTransfer
Index
A
Accessory friction, 308
Adiabatic flame temperature, 100
Air/fuel ratio
definition, 76
oxygen sensor, 353
stoichiometric, 76
Alcohol, 265, 279
Alternative fuels, 274
Antoine’s equation, 75
Aromatics, 265
Atmosphere, standard, 68, 158, 402
Atomization, 163, 173
Auto-ignition, 7, 197, 215
Available energy, 99,104, 372
B
Balance, 19
Bearings, 295
Benz, K., 5
Biodiesel, 285
Blowby, 49, 340
Blowdown, 54, 138
Brake mean effective pressure (bmep), 12
Brake specific fuel consumption (bsfc), 14
C
Carbon monoxide, 85, 243, 358
Carburetor, 163
Carnot, S., 32
Catalytic converter
efficiency, 256
reactions, 255
Cetane index, 283
Cetane number, 219, 280, 284
Charging efficiency, 185
Chemical equilibrium, 84
Choked flow, 133,155,165
Clausius-Clapeyron equation, 75
Clerk, D., 4
Combustion analysis, 354
Combustion diagnostics, 214
Combustion duration, 43, 207
Combustion visualization, 214
Complete expansion, 40
Compression ratio
definition, 10
effects on performance, 124, 373, 383
fuel-air cycle, 108, 128
gas cycle, 35, 38, 40
Compressor map, 155
Compressors, 150
Computational fluid dynamics (CFD), 175
Controls, electronics, 366
Cooling system, 5, 22, 319
Cooperative Fuel Research (CFR) engine, 212
Crevice volume, 246
Crude oils, 262--266
Cumulative energy release fraction, 43
Cycle-to-cycle variations, 198
Cylinder area, 50
Cylinder pressure measurement, 354
Cylinder volume, 17
D
Daimler, G., 5
Delivery ratio, 190
Deposits, 246
Diesel cycle, 7, 36
Diesel engines
combustion, 215
HC emissions, 248
numerical models, 225
particulate matter (PM) emissions, 249
performance, 8, 378
Diesel fuel, 83, 282
Diesel, R., 5
Diffusion coefficient, 390
Dilution tunnel, 364
Direct injection, 7
Discharge coefficient
carburetors, 165
poppet valves, 134
ports, 188
Displacement volume, 10
Distillation, 266
Internal Combustion Engines:Applied Thermosciences, Third Edition. Colin R. Ferguson and Allan T. Kirkpatrick.
c 2016 John Wiley & Sons Ltd. Published 2016 by John Wiley & Sons Ltd.
○
455
456
Index
Drag coefficient, 383
Droplet size, 174
Dual cycle, 38, 128
Dynamometer, 10, 347
E
Efficiency
compressor, 154
mechanical, 11, 372
scavenging, 191
thermal, 15, 113, 127, 322, 383
volumetric, 12, 372
Electric motors, 27
Emission regulation, 251
Emissions testing, 369
Energy balance, 320
Energy release
combustion measurements, 200, 356
compression ignition engines, 222
gas cycles, 45
modeling, 41, 45, 205, 334, 356
spark ignition engines, 204
timing, 53
Engine size, 16, 379
Engine speed, 16, 124, 376
Ensemble average, 181
Enthalpy
formation, 74
vaporization, 73--75
Entropy, 69, 99, 269
Equilibrium
composition, 89
constants, 79, 90
Equivalence ratio
CFR engine, 124
definition, 77
fuel-air cycle, 110--127
mass fraction burned, 206
measurement, 364
Ethanol, 279
Exhaust
analyzers, 364
heat transfer, 338
ideal 4 stroke, 55, 113
manifold, 189
Exhaust gas recirculation (EGR), 206, 254, 354
F
Federal driving schedule, 369
Finite energy release, 41
Fischer-Tropsch reactions, 285
Flame ionization detector (FID), 360
Flame propagation, 201
Flame quenching, 246
Flammability limit, 275
Flow area, 134
Flow bench, 135, 187
Flow coefficient, 135
Flowmeters, 350
Four stroke cycle
definition, 6
exhaust stroke, 55
intake stroke, 57
P-V diagram, 56
Friction
fmep definition, 291
journal bearings, 295
modeling, 294
motoring, 292
oil film, 305
piston and ring, 298
valve train, 306
Fuel-air ratio, 76, 165, 244, 372
Fuel cells, 27
Fuel injection, 166, 353
Fuels
additives, 273
properties, 99, 273, 284, 397
G
Gas constants, 67, 388
Gasoline, 83, 269, 271
Gas turbine, 28
Gibbs free energy, 70, 85
H
HCCI engine, 226
Heat of combustion, 15, 33, 99
Heat transfer
conduction, 327
convection, 327
modeling, 49, 326
measurements, 326
radiation, 339
Helmholtz free energy, 86
Helmholtz resonator, 149
Hybrid electric vehicle, 27
Hydrocarbons
emissions, 245
fuel components, 263
measurement, 359
Hydrogen, 281
I
Ideal gas, 66, 397
Ignition, 7, 198, 215
Ignition delay
compression ignition, 220
Index
spark ignition, 207
Indicated mean effective pressure (imep)
definition, 12
finite energy release model, 47
fuel-air-cycle, 109, 113, 127
Miller cycle, 41
Otto cycle, 35
Indicated specific fuel consumption (ISFC), 14
Indirect injection (IDI), 7, 376, 382
Intake manifold, 147,150
Intake stroke, 57
Internal energy, 67
Isentropic processes, 33, 36, 101, 133
K
Knock
measurements, 206, 368
modeling, 210
457
Mixture mass fraction, 67
Molecular mass, 67, 78
Mole fraction, 67, 389
Motoring mean effective pressure (mmep), 292
N
Naphthenes, 264
Natural gas, 277
Nitrogen oxides
chemical reactions, 85, 235
measurement, 361, 382
rate constants, 236
Nitromethane, 83
Non-methane organic gases (NMOG), 245
Nusselt number, 331, 338
L
Lagrange optimization, 85
Laminar flame speed, 201
Laser Dopper Velocimetry (LDV), 174
Lean NOx trap, 257
Lenior, J., 4
Limited pressure cycle, 38, 125
Low temperature combustion, 225
Lubrication, 312
O
Octane
combustion, 87
number, 5, 212, 280
properties, 101
requirement, 270
Oil, 262, 312
Oil film, 289, 305
Olefins, 264
Otto cycle, 6, 33
Otto, N., 4
Oxygen sensor, 362
M
Mach index, 140
Mach number, 131
Mass blowby, 49
Mass fraction burned, 43, 119
Maximum work, 103
MBT timing, 381
Mean effective pressure
accessory, 291
brake, 12, 158
definitions, 11
friction, 12, 291, 311
indicated, 12, 47, 109, 113, 127
motoring, 292, 380
pumping, 59, 113, 310
Methane, 83, 101
Methane number, 275
Methanol, 83, 101, 279
Microscales
integral, 182
Kolmogorov, 182
Taylor, 182
Midgley, T., 274
Mie scattering, 214
Miller cycle, 39
P
Paraffins, 263
Particle image velocimetry (PIV), 175
Particulates, 249, 363
Part-load performance, 376
PCCI combustion, 227
Penetration layer, 328
Performance maps, 376
Petrov’s equation, 297
Physical constants, 402
Piston
acceleration, 19
force balance, 300
friction, 298
side thrust, 300
skirt, 298
temperature, 329
velocity, 18
wrist pin offset, 300
Piston rings, 298
Piston speed
effect on turbulence, 181
geometric similarity, 16
instantaneous, 18
mean, 10, 376
458
Index
Poppet valve, 20, 132, 307
Power
brake, 10, 372
friction, 11
indicated, 10
road load, 383
Prechamber, 179
Pressure transducers, 354
Propane, 276
Pumping work, 310
Purity, 185
Q
Quality, 76, 411
Quenching, 246
R
Radial engine, 17
Rapid compression machine, 209
Rayleigh scattering, 215
RCCI combustion, 228
Reformulated gasoline (RFG), 272
Residual fraction
fuel-air cycle, 113, 116
gas cycle, 81
measurement, 365
two stroke, 190
valve timing, 147
Reversion, 341
Reynolds equation, 304
Reynolds number, 173, 180, 331
Ricardo, H., 5
Rings, 298
Roots blower, 150
S
Sampling valve, 366
Saturation vapor pressure, 72
Scavenging
analysis, 191
configurations, 186
definition, 8
efficiency, 190
ratio, 190
Second law, 103, 107
Selective catalytic reduction (SCR), 257
Short circuiting, 9, 190
Smoke limit, 249
Soot, 249
Spark ignition
cycles, 6
emissions, 246
performance, 123
Specific fuel consumption, 14, 374
Specific heat
air, 387
ideal gas mixtures, 68--71
motor fuels, 268
Speed of sound, 133, 388
Spray penetration, 173
Squish, 180
Stagnation pressure, 132
Steam engine, 2, 29
Stoichiometry, 76
Stribeck variable, 289
Stroke, 9
Sulphur, 273
Superchargers, 20, 150
Swirl, 175
T
Temperature
cylinder head, 324
piston, 324, 330
Thermal conductivity, 387
Thermal efficiency
Diesel cycle, 36
finite energy release, 46, 120
first law, 15
limited pressure, 38, 127
Miller cycle, 41
Otto cycle, 35
second law, 107
Timing
CFR engine, 124
effect on NOx, 243, 254
spark, 43, 199, 381
valve, 143
Torque, 10
Total hydrocarbons (THC), 245
Trapped air-fuel ratio, 189
Trapping efficiency, 190, 373
Tumble, 175
Tuning, 148
Turbocharger, 20, 150
Turbulence, 5, 180, 254
Turbulence models, 184
Turbulent flame regimes, 202
Two-stroke engines, 8, 185, 315
U
Ultra low sulfur diesel (ULSD), 283
Unit conversions, 401
V
Valve
choked flow, 133
curtain area, 134
Index
discharge coefficient, 134
overlap, 43
poppet, 20, 136
timing, 143
Valve train, 306
Viscosity
air, 387
combustion gas, 331, 328
diesel fuel, 284
oil, 314
Volatility, 270
Volume, 10, 17, 334,379
Volumetric efficiency
definition, 12
fuel-air cycle, 116
gas cycle, 58
speed effect, 12, 146, 149
valve effect, 140, 144
W
Water-gas reaction, 79
Weber number, 173
Wiebe function, 43
Woschni correlation, 333
Z
Zeldovich mechanism, 235
459